SLIDE 1
Which lens spaces embed in S 4 ? Proposition L ( p , q ) , | p | - - PowerPoint PPT Presentation
Which lens spaces embed in S 4 ? Proposition L ( p , q ) , | p | - - PowerPoint PPT Presentation
Embedding 3-manifolds in S 4 Ahmad Issa University of Texas at Austin 1 Embedding manifolds in R n Note: a manifold (of dim < n ) embeds in R n if and only if it embeds in S n . Whitney embedding theorem (1943) A compact smooth n -manifold
SLIDE 2
SLIDE 3
Embedding manifolds in Rn
Note: a manifold (of dim < n) embeds in Rn if and only if it embeds in Sn.
Whitney embedding theorem (1943)
A compact smooth n-manifold can be smoothly embedded in R2n.
Theorem (Hirsch, Wall, Rokhlin 1960’s)
Every 3-manifold smoothly embeds in R5.
Question: (Kirby list 3.20)
Under what conditions does a closed, orientable 3-manifold M smoothly embed in S4? Note that a non-orientable n-manifold cannot smoothly embed in Sn+1.
2
SLIDE 4
Embedding manifolds in Rn
Note: a manifold (of dim < n) embeds in Rn if and only if it embeds in Sn.
Whitney embedding theorem (1943)
A compact smooth n-manifold can be smoothly embedded in R2n.
Theorem (Hirsch, Wall, Rokhlin 1960’s)
Every 3-manifold smoothly embeds in R5.
Question: (Kirby list 3.20)
Under what conditions does a closed, orientable 3-manifold M smoothly embed in S4? Note that a non-orientable n-manifold cannot smoothly embed in Sn+1.
2
SLIDE 5
Embedding manifolds in Rn
Note: a manifold (of dim < n) embeds in Rn if and only if it embeds in Sn.
Whitney embedding theorem (1943)
A compact smooth n-manifold can be smoothly embedded in R2n.
Theorem (Hirsch, Wall, Rokhlin 1960’s)
Every 3-manifold smoothly embeds in R5.
Question: (Kirby list 3.20)
Under what conditions does a closed, orientable 3-manifold M smoothly embed in S4? Note that a non-orientable n-manifold cannot smoothly embed in Sn+1.
2
SLIDE 6
Embedding manifolds in Rn
Note: a manifold (of dim < n) embeds in Rn if and only if it embeds in Sn.
Whitney embedding theorem (1943)
A compact smooth n-manifold can be smoothly embedded in R2n.
Theorem (Hirsch, Wall, Rokhlin 1960’s)
Every 3-manifold smoothly embeds in R5.
Question: (Kirby list 3.20)
Under what conditions does a closed, orientable 3-manifold M smoothly embed in S4? Note that a non-orientable n-manifold cannot smoothly embed in Sn+1.
2
SLIDE 7
Embedding manifolds in Rn
Note: a manifold (of dim < n) embeds in Rn if and only if it embeds in Sn.
Whitney embedding theorem (1943)
A compact smooth n-manifold can be smoothly embedded in R2n.
Theorem (Hirsch, Wall, Rokhlin 1960’s)
Every 3-manifold smoothly embeds in R5.
Question: (Kirby list 3.20)
Under what conditions does a closed, orientable 3-manifold M smoothly embed in S4? Note that a non-orientable n-manifold cannot smoothly embed in Sn+1.
2
SLIDE 8
Embedding 3-manifolds in S4
Simple examples of 3-manifolds which embed: S3, S2 × S1, 3-torus.
Definition
A link L in S3 is strongly slice if L = ∂D, where D ⊂ B4 is a disjoint union of smoothly embedded disks.
Definition
A link L is bipartedly slice if L = L1 ⊔ L2, where L1, L2 are strongly slice.
Fact
The 3-manifold given by 0-surgery on each component of a bipartedly slice link smoothly embeds in S4. Examples:
3
SLIDE 9
Embedding 3-manifolds in S4
Simple examples of 3-manifolds which embed: S3, S2 × S1, 3-torus.
Definition
A link L in S3 is strongly slice if L = ∂D, where D ⊂ B4 is a disjoint union of smoothly embedded disks.
Definition
A link L is bipartedly slice if L = L1 ⊔ L2, where L1, L2 are strongly slice.
Fact
The 3-manifold given by 0-surgery on each component of a bipartedly slice link smoothly embeds in S4. Examples:
3
SLIDE 10
Embedding 3-manifolds in S4
Simple examples of 3-manifolds which embed: S3, S2 × S1, 3-torus.
Definition
A link L in S3 is strongly slice if L = ∂D, where D ⊂ B4 is a disjoint union of smoothly embedded disks.
Definition
A link L is bipartedly slice if L = L1 ⊔ L2, where L1, L2 are strongly slice.
Fact
The 3-manifold given by 0-surgery on each component of a bipartedly slice link smoothly embeds in S4. Examples:
3
SLIDE 11
Embedding 3-manifolds in S4
Simple examples of 3-manifolds which embed: S3, S2 × S1, 3-torus.
Definition
A link L in S3 is strongly slice if L = ∂D, where D ⊂ B4 is a disjoint union of smoothly embedded disks.
Definition
A link L is bipartedly slice if L = L1 ⊔ L2, where L1, L2 are strongly slice.
Fact
The 3-manifold given by 0-surgery on each component of a bipartedly slice link smoothly embeds in S4. Examples:
3
SLIDE 12
Embedding 3-manifolds in S4
Simple examples of 3-manifolds which embed: S3, S2 × S1, 3-torus.
Definition
A link L in S3 is strongly slice if L = ∂D, where D ⊂ B4 is a disjoint union of smoothly embedded disks.
Definition
A link L is bipartedly slice if L = L1 ⊔ L2, where L1, L2 are strongly slice.
Fact
The 3-manifold given by 0-surgery on each component of a bipartedly slice link smoothly embeds in S4. Examples:
L1 S1 x S2
3
SLIDE 13
Embedding 3-manifolds in S4
Simple examples of 3-manifolds which embed: S3, S2 × S1, 3-torus.
Definition
A link L in S3 is strongly slice if L = ∂D, where D ⊂ B4 is a disjoint union of smoothly embedded disks.
Definition
A link L is bipartedly slice if L = L1 ⊔ L2, where L1, L2 are strongly slice.
Fact
The 3-manifold given by 0-surgery on each component of a bipartedly slice link smoothly embeds in S4. Examples:
L1 L2 3-torus
3
SLIDE 14
Embedding 3-manifolds in S4
Simple examples of 3-manifolds which embed: S3, S2 × S1, 3-torus.
Definition
A link L in S3 is strongly slice if L = ∂D, where D ⊂ B4 is a disjoint union of smoothly embedded disks.
Definition
A link L is bipartedly slice if L = L1 ⊔ L2, where L1, L2 are strongly slice.
Fact
The 3-manifold given by 0-surgery on each component of a bipartedly slice link smoothly embeds in S4. Examples:
L1 L2 T2-bundle with monodromy a Dehn twist
3
SLIDE 15
Embedding 3-manifolds in S4
Simple examples of 3-manifolds which embed: S3, S2 × S1, 3-torus.
Definition
A link L in S3 is strongly slice if L = ∂D, where D ⊂ B4 is a disjoint union of smoothly embedded disks.
Definition
A link L is bipartedly slice if L = L1 ⊔ L2, where L1, L2 are strongly slice.
Fact
The 3-manifold given by 0-surgery on each component of a bipartedly slice link smoothly embeds in S4. Examples:
S4 S3
3
SLIDE 16
Embedding 3-manifolds in S4
Simple examples of 3-manifolds which embed: S3, S2 × S1, 3-torus.
Definition
A link L in S3 is strongly slice if L = ∂D, where D ⊂ B4 is a disjoint union of smoothly embedded disks.
Definition
A link L is bipartedly slice if L = L1 ⊔ L2, where L1, L2 are strongly slice.
Fact
The 3-manifold given by 0-surgery on each component of a bipartedly slice link smoothly embeds in S4. Examples:
S4 S3 L1 L2
3
SLIDE 17
Embedding 3-manifolds in S4
Simple examples of 3-manifolds which embed: S3, S2 × S1, 3-torus.
Definition
A link L in S3 is strongly slice if L = ∂D, where D ⊂ B4 is a disjoint union of smoothly embedded disks.
Definition
A link L is bipartedly slice if L = L1 ⊔ L2, where L1, L2 are strongly slice.
Fact
The 3-manifold given by 0-surgery on each component of a bipartedly slice link smoothly embeds in S4. Examples:
S4 S3 slice disk
3
SLIDE 18
Embedding 3-manifolds in S4
Simple examples of 3-manifolds which embed: S3, S2 × S1, 3-torus.
Definition
A link L in S3 is strongly slice if L = ∂D, where D ⊂ B4 is a disjoint union of smoothly embedded disks.
Definition
A link L is bipartedly slice if L = L1 ⊔ L2, where L1, L2 are strongly slice.
Fact
The 3-manifold given by 0-surgery on each component of a bipartedly slice link smoothly embeds in S4. Examples:
S4 S3
attach handle 3
SLIDE 19
Embedding 3-manifolds in S4
Simple examples of 3-manifolds which embed: S3, S2 × S1, 3-torus.
Definition
A link L in S3 is strongly slice if L = ∂D, where D ⊂ B4 is a disjoint union of smoothly embedded disks.
Definition
A link L is bipartedly slice if L = L1 ⊔ L2, where L1, L2 are strongly slice.
Fact
The 3-manifold given by 0-surgery on each component of a bipartedly slice link smoothly embeds in S4. Examples:
S4 S3
0-surgery on link 3
SLIDE 20
Which lens spaces embed in S4?
Proposition
L(p, q), |p| > 1, does not embed in S4. This is a consequence of the following:
Proposition (Hantzsche, 1938)
If M3 embeds in S4 then tor(H1(M)) ∼ = G ⊕ G for some finite abelian group G. Idea: 0 = H2(S4) → H1(M)
∼ =
→ H1(V1) ⊕ H1(V2) → H1(S4) = 0 Notice that H1(L(p, q)) = Z|p| = G ⊕ G for |p| > 1.
Theorem (Epstein 1965, Zeeman)
L(p, q)\˚ B3, p > 1 embeds in S4 if and only if p is odd.
Theorem (Fintushel-Stern, 1985)
If L(p, q)#L(p, q′) embeds in S4, then p is odd and L(p, q) is diffeomorphic to L(p, q′). This was generalised by Donald (2012) to arbitrary connect sums of lens spaces.
4
SLIDE 21
Which lens spaces embed in S4?
Proposition
L(p, q), |p| > 1, does not embed in S4. This is a consequence of the following:
Proposition (Hantzsche, 1938)
If M3 embeds in S4 then tor(H1(M)) ∼ = G ⊕ G for some finite abelian group G. Idea: 0 = H2(S4) → H1(M)
∼ =
→ H1(V1) ⊕ H1(V2) → H1(S4) = 0 Notice that H1(L(p, q)) = Z|p| = G ⊕ G for |p| > 1.
Theorem (Epstein 1965, Zeeman)
L(p, q)\˚ B3, p > 1 embeds in S4 if and only if p is odd.
Theorem (Fintushel-Stern, 1985)
If L(p, q)#L(p, q′) embeds in S4, then p is odd and L(p, q) is diffeomorphic to L(p, q′). This was generalised by Donald (2012) to arbitrary connect sums of lens spaces.
4
SLIDE 22
Which lens spaces embed in S4?
Proposition
L(p, q), |p| > 1, does not embed in S4. This is a consequence of the following:
Proposition (Hantzsche, 1938)
If M3 embeds in S4 then tor(H1(M)) ∼ = G ⊕ G for some finite abelian group G. Idea: 0 = H2(S4) → H1(M)
∼ =
→ H1(V1) ⊕ H1(V2) → H1(S4) = 0 Notice that H1(L(p, q)) = Z|p| = G ⊕ G for |p| > 1.
Theorem (Epstein 1965, Zeeman)
L(p, q)\˚ B3, p > 1 embeds in S4 if and only if p is odd.
Theorem (Fintushel-Stern, 1985)
If L(p, q)#L(p, q′) embeds in S4, then p is odd and L(p, q) is diffeomorphic to L(p, q′). This was generalised by Donald (2012) to arbitrary connect sums of lens spaces.
4
SLIDE 23
Which lens spaces embed in S4?
Proposition
L(p, q), |p| > 1, does not embed in S4. This is a consequence of the following:
Proposition (Hantzsche, 1938)
If M3 embeds in S4 then tor(H1(M)) ∼ = G ⊕ G for some finite abelian group G. Idea: 0 = H2(S4) → H1(M)
∼ =
→ H1(V1) ⊕ H1(V2) → H1(S4) = 0 Notice that H1(L(p, q)) = Z|p| = G ⊕ G for |p| > 1.
Theorem (Epstein 1965, Zeeman)
L(p, q)\˚ B3, p > 1 embeds in S4 if and only if p is odd.
Theorem (Fintushel-Stern, 1985)
If L(p, q)#L(p, q′) embeds in S4, then p is odd and L(p, q) is diffeomorphic to L(p, q′). This was generalised by Donald (2012) to arbitrary connect sums of lens spaces.
4
SLIDE 24
Which lens spaces embed in S4?
Proposition
L(p, q), |p| > 1, does not embed in S4. This is a consequence of the following:
Proposition (Hantzsche, 1938)
If M3 embeds in S4 then tor(H1(M)) ∼ = G ⊕ G for some finite abelian group G. Idea: 0 = H2(S4) → H1(M)
∼ =
→ H1(V1) ⊕ H1(V2) → H1(S4) = 0 Notice that H1(L(p, q)) = Z|p| = G ⊕ G for |p| > 1.
Theorem (Epstein 1965, Zeeman)
L(p, q)\˚ B3, p > 1 embeds in S4 if and only if p is odd.
Theorem (Fintushel-Stern, 1985)
If L(p, q)#L(p, q′) embeds in S4, then p is odd and L(p, q) is diffeomorphic to L(p, q′). This was generalised by Donald (2012) to arbitrary connect sums of lens spaces.
4
SLIDE 25
Which lens spaces embed in S4?
Proposition
L(p, q), |p| > 1, does not embed in S4. This is a consequence of the following:
Proposition (Hantzsche, 1938)
If M3 embeds in S4 then tor(H1(M)) ∼ = G ⊕ G for some finite abelian group G. Idea: 0 = H2(S4) → H1(M)
∼ =
→ H1(V1) ⊕ H1(V2) → H1(S4) = 0 Notice that H1(L(p, q)) = Z|p| = G ⊕ G for |p| > 1.
Theorem (Epstein 1965, Zeeman)
L(p, q)\˚ B3, p > 1 embeds in S4 if and only if p is odd.
Theorem (Fintushel-Stern, 1985)
If L(p, q)#L(p, q′) embeds in S4, then p is odd and L(p, q) is diffeomorphic to L(p, q′). This was generalised by Donald (2012) to arbitrary connect sums of lens spaces.
4
SLIDE 26
Which lens spaces embed in S4?
Proposition
L(p, q), |p| > 1, does not embed in S4. This is a consequence of the following:
Proposition (Hantzsche, 1938)
If M3 embeds in S4 then tor(H1(M)) ∼ = G ⊕ G for some finite abelian group G. Idea: 0 = H2(S4) → H1(M)
∼ =
→ H1(V1) ⊕ H1(V2) → H1(S4) = 0 Notice that H1(L(p, q)) = Z|p| = G ⊕ G for |p| > 1.
Theorem (Epstein 1965, Zeeman)
L(p, q)\˚ B3, p > 1 embeds in S4 if and only if p is odd.
Theorem (Fintushel-Stern, 1985)
If L(p, q)#L(p, q′) embeds in S4, then p is odd and L(p, q) is diffeomorphic to L(p, q′). This was generalised by Donald (2012) to arbitrary connect sums of lens spaces.
4
SLIDE 27
Which lens spaces embed in S4?
Proposition
L(p, q), |p| > 1, does not embed in S4. This is a consequence of the following:
Proposition (Hantzsche, 1938)
If M3 embeds in S4 then tor(H1(M)) ∼ = G ⊕ G for some finite abelian group G. Idea: 0 = H2(S4) → H1(M)
∼ =
→ H1(V1) ⊕ H1(V2) → H1(S4) = 0 Notice that H1(L(p, q)) = Z|p| = G ⊕ G for |p| > 1.
Theorem (Epstein 1965, Zeeman)
L(p, q)\˚ B3, p > 1 embeds in S4 if and only if p is odd.
Theorem (Fintushel-Stern, 1985)
If L(p, q)#L(p, q′) embeds in S4, then p is odd and L(p, q) is diffeomorphic to L(p, q′). This was generalised by Donald (2012) to arbitrary connect sums of lens spaces.
4
SLIDE 28
Which lens spaces embed in S4?
Proposition
L(p, q), |p| > 1, does not embed in S4. This is a consequence of the following:
Proposition (Hantzsche, 1938)
If M3 embeds in S4 then tor(H1(M)) ∼ = G ⊕ G for some finite abelian group G. Idea: 0 = H2(S4) → H1(M)
∼ =
→ H1(V1) ⊕ H1(V2) → H1(S4) = 0 Notice that H1(L(p, q)) = Z|p| = G ⊕ G for |p| > 1.
Theorem (Epstein 1965, Zeeman)
L(p, q)\˚ B3, p > 1 embeds in S4 if and only if p is odd.
Theorem (Fintushel-Stern, 1985)
If L(p, q)#L(p, q′) embeds in S4, then p is odd and L(p, q) is diffeomorphic to L(p, q′). This was generalised by Donald (2012) to arbitrary connect sums of lens spaces.
4
SLIDE 29
Embedding homology spheres
Theorem (Freedman, 1982)
Every Z-homology sphere topologically locally flatly embeds in S4. The question of smooth embeddings is much more subtle. For example: Σ(2, 3, 5) does not smoothly embed in S4:
Lemma
If a Z-homology sphere M3 embeds in S4 then it separates S4 into two Z-homology B4’s V1 and V2. In particular M bounds an acyclic manifold (a Z-homology B4). Sketch: Hi+1(S4) → Hi(M) → Hi(V1) ⊕ Hi(V2) → Hi(S4).
Theorem (Rokhlin, 1952)
If X 4 is a closed smooth 4-mfd with H1(X 4) = 0 and even intersection form then σ(X)/8 = 0 (mod 2).
◮ Σ(2, 3, 5) bounds the E8 plumbing W 4. ◮ If Σ(2, 3, 5) embeds in S4 then V1 ∪∂ W satisfies conditions of
Rokhlin’s theorem.
◮ However, σ(V1 ∪ W )/8 = −8/8 = 1 (mod 2), a contradiction. 5
SLIDE 30
Embedding homology spheres
Theorem (Freedman, 1982)
Every Z-homology sphere topologically locally flatly embeds in S4. The question of smooth embeddings is much more subtle. For example: Σ(2, 3, 5) does not smoothly embed in S4:
Lemma
If a Z-homology sphere M3 embeds in S4 then it separates S4 into two Z-homology B4’s V1 and V2. In particular M bounds an acyclic manifold (a Z-homology B4). Sketch: Hi+1(S4) → Hi(M) → Hi(V1) ⊕ Hi(V2) → Hi(S4).
Theorem (Rokhlin, 1952)
If X 4 is a closed smooth 4-mfd with H1(X 4) = 0 and even intersection form then σ(X)/8 = 0 (mod 2).
◮ Σ(2, 3, 5) bounds the E8 plumbing W 4. ◮ If Σ(2, 3, 5) embeds in S4 then V1 ∪∂ W satisfies conditions of
Rokhlin’s theorem.
◮ However, σ(V1 ∪ W )/8 = −8/8 = 1 (mod 2), a contradiction. 5
SLIDE 31
Embedding homology spheres
Theorem (Freedman, 1982)
Every Z-homology sphere topologically locally flatly embeds in S4. The question of smooth embeddings is much more subtle. For example: Σ(2, 3, 5) does not smoothly embed in S4:
Lemma
If a Z-homology sphere M3 embeds in S4 then it separates S4 into two Z-homology B4’s V1 and V2. In particular M bounds an acyclic manifold (a Z-homology B4). Sketch: Hi+1(S4) → Hi(M) → Hi(V1) ⊕ Hi(V2) → Hi(S4).
Theorem (Rokhlin, 1952)
If X 4 is a closed smooth 4-mfd with H1(X 4) = 0 and even intersection form then σ(X)/8 = 0 (mod 2).
◮ Σ(2, 3, 5) bounds the E8 plumbing W 4. ◮ If Σ(2, 3, 5) embeds in S4 then V1 ∪∂ W satisfies conditions of
Rokhlin’s theorem.
◮ However, σ(V1 ∪ W )/8 = −8/8 = 1 (mod 2), a contradiction. 5
SLIDE 32
Embedding homology spheres
Theorem (Freedman, 1982)
Every Z-homology sphere topologically locally flatly embeds in S4. The question of smooth embeddings is much more subtle. For example: Σ(2, 3, 5) does not smoothly embed in S4:
Lemma
If a Z-homology sphere M3 embeds in S4 then it separates S4 into two Z-homology B4’s V1 and V2. In particular M bounds an acyclic manifold (a Z-homology B4). Sketch: Hi+1(S4) → Hi(M) → Hi(V1) ⊕ Hi(V2) → Hi(S4).
Theorem (Rokhlin, 1952)
If X 4 is a closed smooth 4-mfd with H1(X 4) = 0 and even intersection form then σ(X)/8 = 0 (mod 2).
◮ Σ(2, 3, 5) bounds the E8 plumbing W 4. ◮ If Σ(2, 3, 5) embeds in S4 then V1 ∪∂ W satisfies conditions of
Rokhlin’s theorem.
◮ However, σ(V1 ∪ W )/8 = −8/8 = 1 (mod 2), a contradiction. 5
SLIDE 33
Embedding homology spheres
Theorem (Freedman, 1982)
Every Z-homology sphere topologically locally flatly embeds in S4. The question of smooth embeddings is much more subtle. For example: Σ(2, 3, 5) does not smoothly embed in S4:
Lemma
If a Z-homology sphere M3 embeds in S4 then it separates S4 into two Z-homology B4’s V1 and V2. In particular M bounds an acyclic manifold (a Z-homology B4). Sketch: Hi+1(S4) → Hi(M) → Hi(V1) ⊕ Hi(V2) → Hi(S4).
Theorem (Rokhlin, 1952)
If X 4 is a closed smooth 4-mfd with H1(X 4) = 0 and even intersection form then σ(X)/8 = 0 (mod 2).
◮ Σ(2, 3, 5) bounds the E8 plumbing W 4. ◮ If Σ(2, 3, 5) embeds in S4 then V1 ∪∂ W satisfies conditions of
Rokhlin’s theorem.
◮ However, σ(V1 ∪ W )/8 = −8/8 = 1 (mod 2), a contradiction. 5
SLIDE 34
Embedding homology spheres
Theorem (Freedman, 1982)
Every Z-homology sphere topologically locally flatly embeds in S4. The question of smooth embeddings is much more subtle. For example: Σ(2, 3, 5) does not smoothly embed in S4:
Lemma
If a Z-homology sphere M3 embeds in S4 then it separates S4 into two Z-homology B4’s V1 and V2. In particular M bounds an acyclic manifold (a Z-homology B4). Sketch: Hi+1(S4) → Hi(M) → Hi(V1) ⊕ Hi(V2) → Hi(S4).
Theorem (Rokhlin, 1952)
If X 4 is a closed smooth 4-mfd with H1(X 4) = 0 and even intersection form then σ(X)/8 = 0 (mod 2).
◮ Σ(2, 3, 5) bounds the E8 plumbing W 4. ◮ If Σ(2, 3, 5) embeds in S4 then V1 ∪∂ W satisfies conditions of
Rokhlin’s theorem.
◮ However, σ(V1 ∪ W )/8 = −8/8 = 1 (mod 2), a contradiction. 5
SLIDE 35
Embedding homology spheres
Theorem (Freedman, 1982)
Every Z-homology sphere topologically locally flatly embeds in S4. The question of smooth embeddings is much more subtle. For example: Σ(2, 3, 5) does not smoothly embed in S4:
Lemma
If a Z-homology sphere M3 embeds in S4 then it separates S4 into two Z-homology B4’s V1 and V2. In particular M bounds an acyclic manifold (a Z-homology B4). Sketch: Hi+1(S4) → Hi(M) → Hi(V1) ⊕ Hi(V2) → Hi(S4).
Theorem (Rokhlin, 1952)
If X 4 is a closed smooth 4-mfd with H1(X 4) = 0 and even intersection form then σ(X)/8 = 0 (mod 2).
◮ Σ(2, 3, 5) bounds the E8 plumbing W 4. ◮ If Σ(2, 3, 5) embeds in S4 then V1 ∪∂ W satisfies conditions of
Rokhlin’s theorem.
◮ However, σ(V1 ∪ W )/8 = −8/8 = 1 (mod 2), a contradiction. 5
SLIDE 36
Embedding homology spheres
Theorem (Freedman, 1982)
Every Z-homology sphere topologically locally flatly embeds in S4. The question of smooth embeddings is much more subtle. For example: Σ(2, 3, 5) does not smoothly embed in S4:
Lemma
If a Z-homology sphere M3 embeds in S4 then it separates S4 into two Z-homology B4’s V1 and V2. In particular M bounds an acyclic manifold (a Z-homology B4). Sketch: Hi+1(S4) → Hi(M) → Hi(V1) ⊕ Hi(V2) → Hi(S4).
Theorem (Rokhlin, 1952)
If X 4 is a closed smooth 4-mfd with H1(X 4) = 0 and even intersection form then σ(X)/8 = 0 (mod 2).
◮ Σ(2, 3, 5) bounds the E8 plumbing W 4. ◮ If Σ(2, 3, 5) embeds in S4 then V1 ∪∂ W satisfies conditions of
Rokhlin’s theorem.
◮ However, σ(V1 ∪ W )/8 = −8/8 = 1 (mod 2), a contradiction. 5
SLIDE 37
Embedding homology spheres
Theorem (Freedman, 1982)
Every Z-homology sphere topologically locally flatly embeds in S4. The question of smooth embeddings is much more subtle. For example: Σ(2, 3, 5) does not smoothly embed in S4:
Lemma
If a Z-homology sphere M3 embeds in S4 then it separates S4 into two Z-homology B4’s V1 and V2. In particular M bounds an acyclic manifold (a Z-homology B4). Sketch: Hi+1(S4) → Hi(M) → Hi(V1) ⊕ Hi(V2) → Hi(S4).
Theorem (Rokhlin, 1952)
If X 4 is a closed smooth 4-mfd with H1(X 4) = 0 and even intersection form then σ(X)/8 = 0 (mod 2).
◮ Σ(2, 3, 5) bounds the E8 plumbing W 4. ◮ If Σ(2, 3, 5) embeds in S4 then V1 ∪∂ W satisfies conditions of
Rokhlin’s theorem.
◮ However, σ(V1 ∪ W )/8 = −8/8 = 1 (mod 2), a contradiction. 5
SLIDE 38
Embedding homology spheres
Theorem (Freedman, 1982)
Every Z-homology sphere topologically locally flatly embeds in S4. The question of smooth embeddings is much more subtle. For example: Σ(2, 3, 5) does not smoothly embed in S4:
Lemma
If a Z-homology sphere M3 embeds in S4 then it separates S4 into two Z-homology B4’s V1 and V2. In particular M bounds an acyclic manifold (a Z-homology B4). Sketch: Hi+1(S4) → Hi(M) → Hi(V1) ⊕ Hi(V2) → Hi(S4).
Theorem (Rokhlin, 1952)
If X 4 is a closed smooth 4-mfd with H1(X 4) = 0 and even intersection form then σ(X)/8 = 0 (mod 2).
◮ Σ(2, 3, 5) bounds the E8 plumbing W 4. ◮ If Σ(2, 3, 5) embeds in S4 then V1 ∪∂ W satisfies conditions of
Rokhlin’s theorem.
◮ However, σ(V1 ∪ W )/8 = −8/8 = 1 (mod 2), a contradiction. 5
SLIDE 39
Embedding homology spheres
Theorem (Freedman, 1982)
Every Z-homology sphere topologically locally flatly embeds in S4. The question of smooth embeddings is much more subtle. For example: Σ(2, 3, 5) does not smoothly embed in S4:
Lemma
If a Z-homology sphere M3 embeds in S4 then it separates S4 into two Z-homology B4’s V1 and V2. In particular M bounds an acyclic manifold (a Z-homology B4). Sketch: Hi+1(S4) → Hi(M) → Hi(V1) ⊕ Hi(V2) → Hi(S4).
Theorem (Rokhlin, 1952)
If X 4 is a closed smooth 4-mfd with H1(X 4) = 0 and even intersection form then σ(X)/8 = 0 (mod 2).
◮ Σ(2, 3, 5) bounds the E8 plumbing W 4. ◮ If Σ(2, 3, 5) embeds in S4 then V1 ∪∂ W satisfies conditions of
Rokhlin’s theorem.
◮ However, σ(V1 ∪ W )/8 = −8/8 = 1 (mod 2), a contradiction. 5
SLIDE 40
Obstructing Seifert fibered homology spheres
From this point onwards all embeddings are smooth. Let M = Σ(a1, . . . , an), ai > 1 pairwise coprime be a Seifert fibered ZHS. There are two “main” obstructions to M bounding a Z-homology ball:
◮ The Neumann-Siebenmann invariant µ(M)
◮ a spin Z-homology cobordism invariant, ◮ lifts the Rokhlin invariant.
◮ The d-invariant d(M) of Ozsvath-Szabo, a spinc Q-homology
cobordism invariant. Example:
◮ µ(Σ(2, 3, 7)) = 1, d(Σ(2, 3, 7)) = 0. ◮ µ(Σ(3, 5, 7)) = 0, d(Σ(3, 5, 7)) = 2. ◮ So Σ(2, 3, 7) and Σ(3, 5, 7) don’t embed in S4. 6
SLIDE 41
Obstructing Seifert fibered homology spheres
From this point onwards all embeddings are smooth. Let M = Σ(a1, . . . , an), ai > 1 pairwise coprime be a Seifert fibered ZHS. There are two “main” obstructions to M bounding a Z-homology ball:
◮ The Neumann-Siebenmann invariant µ(M)
◮ a spin Z-homology cobordism invariant, ◮ lifts the Rokhlin invariant.
◮ The d-invariant d(M) of Ozsvath-Szabo, a spinc Q-homology
cobordism invariant. Example:
◮ µ(Σ(2, 3, 7)) = 1, d(Σ(2, 3, 7)) = 0. ◮ µ(Σ(3, 5, 7)) = 0, d(Σ(3, 5, 7)) = 2. ◮ So Σ(2, 3, 7) and Σ(3, 5, 7) don’t embed in S4. 6
SLIDE 42
Obstructing Seifert fibered homology spheres
From this point onwards all embeddings are smooth. Let M = Σ(a1, . . . , an), ai > 1 pairwise coprime be a Seifert fibered ZHS. There are two “main” obstructions to M bounding a Z-homology ball:
◮ The Neumann-Siebenmann invariant µ(M)
◮ a spin Z-homology cobordism invariant, ◮ lifts the Rokhlin invariant.
◮ The d-invariant d(M) of Ozsvath-Szabo, a spinc Q-homology
cobordism invariant. Example:
◮ µ(Σ(2, 3, 7)) = 1, d(Σ(2, 3, 7)) = 0. ◮ µ(Σ(3, 5, 7)) = 0, d(Σ(3, 5, 7)) = 2. ◮ So Σ(2, 3, 7) and Σ(3, 5, 7) don’t embed in S4. 6
SLIDE 43
Obstructing Seifert fibered homology spheres
From this point onwards all embeddings are smooth. Let M = Σ(a1, . . . , an), ai > 1 pairwise coprime be a Seifert fibered ZHS. There are two “main” obstructions to M bounding a Z-homology ball:
◮ The Neumann-Siebenmann invariant µ(M)
◮ a spin Z-homology cobordism invariant, ◮ lifts the Rokhlin invariant.
◮ The d-invariant d(M) of Ozsvath-Szabo, a spinc Q-homology
cobordism invariant. Example:
◮ µ(Σ(2, 3, 7)) = 1, d(Σ(2, 3, 7)) = 0. ◮ µ(Σ(3, 5, 7)) = 0, d(Σ(3, 5, 7)) = 2. ◮ So Σ(2, 3, 7) and Σ(3, 5, 7) don’t embed in S4. 6
SLIDE 44
Obstructing Seifert fibered homology spheres
From this point onwards all embeddings are smooth. Let M = Σ(a1, . . . , an), ai > 1 pairwise coprime be a Seifert fibered ZHS. There are two “main” obstructions to M bounding a Z-homology ball:
◮ The Neumann-Siebenmann invariant µ(M)
◮ a spin Z-homology cobordism invariant, ◮ lifts the Rokhlin invariant.
◮ The d-invariant d(M) of Ozsvath-Szabo, a spinc Q-homology
cobordism invariant. Example:
◮ µ(Σ(2, 3, 7)) = 1, d(Σ(2, 3, 7)) = 0. ◮ µ(Σ(3, 5, 7)) = 0, d(Σ(3, 5, 7)) = 2. ◮ So Σ(2, 3, 7) and Σ(3, 5, 7) don’t embed in S4. 6
SLIDE 45
Obstructing Seifert fibered homology spheres
From this point onwards all embeddings are smooth. Let M = Σ(a1, . . . , an), ai > 1 pairwise coprime be a Seifert fibered ZHS. There are two “main” obstructions to M bounding a Z-homology ball:
◮ The Neumann-Siebenmann invariant µ(M)
◮ a spin Z-homology cobordism invariant, ◮ lifts the Rokhlin invariant.
◮ The d-invariant d(M) of Ozsvath-Szabo, a spinc Q-homology
cobordism invariant. Example:
◮ µ(Σ(2, 3, 7)) = 1, d(Σ(2, 3, 7)) = 0. ◮ µ(Σ(3, 5, 7)) = 0, d(Σ(3, 5, 7)) = 2. ◮ So Σ(2, 3, 7) and Σ(3, 5, 7) don’t embed in S4. 6
SLIDE 46
Obstructing Seifert fibered homology spheres
From this point onwards all embeddings are smooth. Let M = Σ(a1, . . . , an), ai > 1 pairwise coprime be a Seifert fibered ZHS. There are two “main” obstructions to M bounding a Z-homology ball:
◮ The Neumann-Siebenmann invariant µ(M)
◮ a spin Z-homology cobordism invariant, ◮ lifts the Rokhlin invariant.
◮ The d-invariant d(M) of Ozsvath-Szabo, a spinc Q-homology
cobordism invariant. Example:
◮ µ(Σ(2, 3, 7)) = 1, d(Σ(2, 3, 7)) = 0. ◮ µ(Σ(3, 5, 7)) = 0, d(Σ(3, 5, 7)) = 2. ◮ So Σ(2, 3, 7) and Σ(3, 5, 7) don’t embed in S4. 6
SLIDE 47
Obstructing Seifert fibered homology spheres
From this point onwards all embeddings are smooth. Let M = Σ(a1, . . . , an), ai > 1 pairwise coprime be a Seifert fibered ZHS. There are two “main” obstructions to M bounding a Z-homology ball:
◮ The Neumann-Siebenmann invariant µ(M)
◮ a spin Z-homology cobordism invariant, ◮ lifts the Rokhlin invariant.
◮ The d-invariant d(M) of Ozsvath-Szabo, a spinc Q-homology
cobordism invariant. Example:
◮ µ(Σ(2, 3, 7)) = 1, d(Σ(2, 3, 7)) = 0. ◮ µ(Σ(3, 5, 7)) = 0, d(Σ(3, 5, 7)) = 2. ◮ So Σ(2, 3, 7) and Σ(3, 5, 7) don’t embed in S4. 6
SLIDE 48
Obstructing Seifert fibered homology spheres
From this point onwards all embeddings are smooth. Let M = Σ(a1, . . . , an), ai > 1 pairwise coprime be a Seifert fibered ZHS. There are two “main” obstructions to M bounding a Z-homology ball:
◮ The Neumann-Siebenmann invariant µ(M)
◮ a spin Z-homology cobordism invariant, ◮ lifts the Rokhlin invariant.
◮ The d-invariant d(M) of Ozsvath-Szabo, a spinc Q-homology
cobordism invariant. Example:
◮ µ(Σ(2, 3, 7)) = 1, d(Σ(2, 3, 7)) = 0. ◮ µ(Σ(3, 5, 7)) = 0, d(Σ(3, 5, 7)) = 2. ◮ So Σ(2, 3, 7) and Σ(3, 5, 7) don’t embed in S4. 6
SLIDE 49
Obstructing Seifert fibered homology spheres
From this point onwards all embeddings are smooth. Let M = Σ(a1, . . . , an), ai > 1 pairwise coprime be a Seifert fibered ZHS. There are two “main” obstructions to M bounding a Z-homology ball:
◮ The Neumann-Siebenmann invariant µ(M)
◮ a spin Z-homology cobordism invariant, ◮ lifts the Rokhlin invariant.
◮ The d-invariant d(M) of Ozsvath-Szabo, a spinc Q-homology
cobordism invariant. Example:
◮ µ(Σ(2, 3, 7)) = 1, d(Σ(2, 3, 7)) = 0. ◮ µ(Σ(3, 5, 7)) = 0, d(Σ(3, 5, 7)) = 2. ◮ So Σ(2, 3, 7) and Σ(3, 5, 7) don’t embed in S4. 6
SLIDE 50
Obstructing Seifert fibered homology spheres
From this point onwards all embeddings are smooth. Let M = Σ(a1, . . . , an), ai > 1 pairwise coprime be a Seifert fibered ZHS. There are two “main” obstructions to M bounding a Z-homology ball:
◮ The Neumann-Siebenmann invariant µ(M)
◮ a spin Z-homology cobordism invariant, ◮ lifts the Rokhlin invariant.
◮ The d-invariant d(M) of Ozsvath-Szabo, a spinc Q-homology
cobordism invariant. Example:
◮ µ(Σ(2, 3, 7)) = 1, d(Σ(2, 3, 7)) = 0. ◮ µ(Σ(3, 5, 7)) = 0, d(Σ(3, 5, 7)) = 2. ◮ So Σ(2, 3, 7) and Σ(3, 5, 7) don’t embed in S4. 6
SLIDE 51
For M a SFHS, if µ(M) = d(M) = 0 then the following obstructions vanish:
◮ Fintushel-Stern’s R-invariant (at least for 3 singular fibers,
Lecuona-Lisca).
◮ Donaldson’s theorem (follows from Elkies, Ozsvath-Szabo) ◮ Manolescu’s α, β, γ invariants coming from Pin(2)-equivariant SWFH
(Stoffregen).
◮ Stoffregen’s SWFHconn invariant coming from Pin(2)-equivariant
SWFH (Stoffregen).
◮ (Conjecturally) d and d invariants of Manolescu-Hendricks coming
from involutive Heegaard-Floer homology. It may be possible that the Seiberg-Witten equations don’t see any further
- bstructions to bounding an acyclic manifold.
7
SLIDE 52
For M a SFHS, if µ(M) = d(M) = 0 then the following obstructions vanish:
◮ Fintushel-Stern’s R-invariant (at least for 3 singular fibers,
Lecuona-Lisca).
◮ Donaldson’s theorem (follows from Elkies, Ozsvath-Szabo) ◮ Manolescu’s α, β, γ invariants coming from Pin(2)-equivariant SWFH
(Stoffregen).
◮ Stoffregen’s SWFHconn invariant coming from Pin(2)-equivariant
SWFH (Stoffregen).
◮ (Conjecturally) d and d invariants of Manolescu-Hendricks coming
from involutive Heegaard-Floer homology. It may be possible that the Seiberg-Witten equations don’t see any further
- bstructions to bounding an acyclic manifold.
7
SLIDE 53
For M a SFHS, if µ(M) = d(M) = 0 then the following obstructions vanish:
◮ Fintushel-Stern’s R-invariant (at least for 3 singular fibers,
Lecuona-Lisca).
◮ Donaldson’s theorem (follows from Elkies, Ozsvath-Szabo) ◮ Manolescu’s α, β, γ invariants coming from Pin(2)-equivariant SWFH
(Stoffregen).
◮ Stoffregen’s SWFHconn invariant coming from Pin(2)-equivariant
SWFH (Stoffregen).
◮ (Conjecturally) d and d invariants of Manolescu-Hendricks coming
from involutive Heegaard-Floer homology. It may be possible that the Seiberg-Witten equations don’t see any further
- bstructions to bounding an acyclic manifold.
7
SLIDE 54
For M a SFHS, if µ(M) = d(M) = 0 then the following obstructions vanish:
◮ Fintushel-Stern’s R-invariant (at least for 3 singular fibers,
Lecuona-Lisca).
◮ Donaldson’s theorem (follows from Elkies, Ozsvath-Szabo) ◮ Manolescu’s α, β, γ invariants coming from Pin(2)-equivariant SWFH
(Stoffregen).
◮ Stoffregen’s SWFHconn invariant coming from Pin(2)-equivariant
SWFH (Stoffregen).
◮ (Conjecturally) d and d invariants of Manolescu-Hendricks coming
from involutive Heegaard-Floer homology. It may be possible that the Seiberg-Witten equations don’t see any further
- bstructions to bounding an acyclic manifold.
7
SLIDE 55
For M a SFHS, if µ(M) = d(M) = 0 then the following obstructions vanish:
◮ Fintushel-Stern’s R-invariant (at least for 3 singular fibers,
Lecuona-Lisca).
◮ Donaldson’s theorem (follows from Elkies, Ozsvath-Szabo) ◮ Manolescu’s α, β, γ invariants coming from Pin(2)-equivariant SWFH
(Stoffregen).
◮ Stoffregen’s SWFHconn invariant coming from Pin(2)-equivariant
SWFH (Stoffregen).
◮ (Conjecturally) d and d invariants of Manolescu-Hendricks coming
from involutive Heegaard-Floer homology. It may be possible that the Seiberg-Witten equations don’t see any further
- bstructions to bounding an acyclic manifold.
7
SLIDE 56
For M a SFHS, if µ(M) = d(M) = 0 then the following obstructions vanish:
◮ Fintushel-Stern’s R-invariant (at least for 3 singular fibers,
Lecuona-Lisca).
◮ Donaldson’s theorem (follows from Elkies, Ozsvath-Szabo) ◮ Manolescu’s α, β, γ invariants coming from Pin(2)-equivariant SWFH
(Stoffregen).
◮ Stoffregen’s SWFHconn invariant coming from Pin(2)-equivariant
SWFH (Stoffregen).
◮ (Conjecturally) d and d invariants of Manolescu-Hendricks coming
from involutive Heegaard-Floer homology. It may be possible that the Seiberg-Witten equations don’t see any further
- bstructions to bounding an acyclic manifold.
7
SLIDE 57
For M a SFHS, if µ(M) = d(M) = 0 then the following obstructions vanish:
◮ Fintushel-Stern’s R-invariant (at least for 3 singular fibers,
Lecuona-Lisca).
◮ Donaldson’s theorem (follows from Elkies, Ozsvath-Szabo) ◮ Manolescu’s α, β, γ invariants coming from Pin(2)-equivariant SWFH
(Stoffregen).
◮ Stoffregen’s SWFHconn invariant coming from Pin(2)-equivariant
SWFH (Stoffregen).
◮ (Conjecturally) d and d invariants of Manolescu-Hendricks coming
from involutive Heegaard-Floer homology. It may be possible that the Seiberg-Witten equations don’t see any further
- bstructions to bounding an acyclic manifold.
7
SLIDE 58
Mazur manifolds
Theorem (Akbulut-Kirby, 1978)
Σ(3, 4, 5) embeds in S4. Proof:
◮ Σ(3, 4, 5) is the boundary of W 4 pictured ◮ W 4 is a Mazur manifold, i.e. a contractible 4-mfld built from a 0-h, a
1-h and a 2-h.
◮ Claim: The double DW of a Mazur manifold W is S4, hence W
embeds in S4. Proof of claim: (Mazur, 1960)
◮ DW = W ∪∂ (−W ) = ∂(W × [0, 1]) ◮ W × [0, 1] is a 5-manifold built from a 0-h, 1-h, 2-h. ◮ 2-h attached along knot γ in ∂(0-h ∪ 1-h) = S3 × S1. ◮ Can unknot γ, so γ isotopic to pt × S1 ⊂ S3 × S1. ◮ Hence, can cancel 1-h and 2-h, so DW = ∂(0-h) = S4. 8
SLIDE 59
Mazur manifolds
Theorem (Akbulut-Kirby, 1978)
Σ(3, 4, 5) embeds in S4. Proof:
◮ Σ(3, 4, 5) is the boundary of W 4 pictured ◮ W 4 is a Mazur manifold, i.e. a contractible 4-mfld built from a 0-h, a
1-h and a 2-h.
◮ Claim: The double DW of a Mazur manifold W is S4, hence W
embeds in S4. Proof of claim: (Mazur, 1960)
◮ DW = W ∪∂ (−W ) = ∂(W × [0, 1]) ◮ W × [0, 1] is a 5-manifold built from a 0-h, 1-h, 2-h. ◮ 2-h attached along knot γ in ∂(0-h ∪ 1-h) = S3 × S1. ◮ Can unknot γ, so γ isotopic to pt × S1 ⊂ S3 × S1. ◮ Hence, can cancel 1-h and 2-h, so DW = ∂(0-h) = S4. 8
SLIDE 60
Mazur manifolds
Theorem (Akbulut-Kirby, 1978)
Σ(3, 4, 5) embeds in S4.
4
Proof:
◮ Σ(3, 4, 5) is the boundary of W 4 pictured ◮ W 4 is a Mazur manifold, i.e. a contractible 4-mfld built from a 0-h, a
1-h and a 2-h.
◮ Claim: The double DW of a Mazur manifold W is S4, hence W
embeds in S4. Proof of claim: (Mazur, 1960)
◮ DW = W ∪∂ (−W ) = ∂(W × [0, 1]) ◮ W × [0, 1] is a 5-manifold built from a 0-h, 1-h, 2-h. ◮ 2-h attached along knot γ in ∂(0-h ∪ 1-h) = S3 × S1. ◮ Can unknot γ, so γ isotopic to pt × S1 ⊂ S3 × S1. ◮ Hence, can cancel 1-h and 2-h, so DW = ∂(0-h) = S4. 8
SLIDE 61
Mazur manifolds
Theorem (Akbulut-Kirby, 1978)
Σ(3, 4, 5) embeds in S4.
4
Proof:
◮ Σ(3, 4, 5) is the boundary of W 4 pictured ◮ W 4 is a Mazur manifold, i.e. a contractible 4-mfld built from a 0-h, a
1-h and a 2-h.
◮ Claim: The double DW of a Mazur manifold W is S4, hence W
embeds in S4. Proof of claim: (Mazur, 1960)
◮ DW = W ∪∂ (−W ) = ∂(W × [0, 1]) ◮ W × [0, 1] is a 5-manifold built from a 0-h, 1-h, 2-h. ◮ 2-h attached along knot γ in ∂(0-h ∪ 1-h) = S3 × S1. ◮ Can unknot γ, so γ isotopic to pt × S1 ⊂ S3 × S1. ◮ Hence, can cancel 1-h and 2-h, so DW = ∂(0-h) = S4. 8
SLIDE 62
Mazur manifolds
Theorem (Akbulut-Kirby, 1978)
Σ(3, 4, 5) embeds in S4.
4
Proof:
◮ Σ(3, 4, 5) is the boundary of W 4 pictured ◮ W 4 is a Mazur manifold, i.e. a contractible 4-mfld built from a 0-h, a
1-h and a 2-h.
◮ Claim: The double DW of a Mazur manifold W is S4, hence W
embeds in S4. Proof of claim: (Mazur, 1960)
◮ DW = W ∪∂ (−W ) = ∂(W × [0, 1]) ◮ W × [0, 1] is a 5-manifold built from a 0-h, 1-h, 2-h. ◮ 2-h attached along knot γ in ∂(0-h ∪ 1-h) = S3 × S1. ◮ Can unknot γ, so γ isotopic to pt × S1 ⊂ S3 × S1. ◮ Hence, can cancel 1-h and 2-h, so DW = ∂(0-h) = S4. 8
SLIDE 63
Mazur manifolds
Theorem (Akbulut-Kirby, 1978)
Σ(3, 4, 5) embeds in S4.
4
Proof:
◮ Σ(3, 4, 5) is the boundary of W 4 pictured ◮ W 4 is a Mazur manifold, i.e. a contractible 4-mfld built from a 0-h, a
1-h and a 2-h.
◮ Claim: The double DW of a Mazur manifold W is S4, hence W
embeds in S4. Proof of claim: (Mazur, 1960)
◮ DW = W ∪∂ (−W ) = ∂(W × [0, 1]) ◮ W × [0, 1] is a 5-manifold built from a 0-h, 1-h, 2-h. ◮ 2-h attached along knot γ in ∂(0-h ∪ 1-h) = S3 × S1. ◮ Can unknot γ, so γ isotopic to pt × S1 ⊂ S3 × S1. ◮ Hence, can cancel 1-h and 2-h, so DW = ∂(0-h) = S4. 8
SLIDE 64
Mazur manifolds
Theorem (Akbulut-Kirby, 1978)
Σ(3, 4, 5) embeds in S4.
4
Proof:
◮ Σ(3, 4, 5) is the boundary of W 4 pictured ◮ W 4 is a Mazur manifold, i.e. a contractible 4-mfld built from a 0-h, a
1-h and a 2-h.
◮ Claim: The double DW of a Mazur manifold W is S4, hence W
embeds in S4. Proof of claim: (Mazur, 1960)
◮ DW = W ∪∂ (−W ) = ∂(W × [0, 1]) ◮ W × [0, 1] is a 5-manifold built from a 0-h, 1-h, 2-h. ◮ 2-h attached along knot γ in ∂(0-h ∪ 1-h) = S3 × S1. ◮ Can unknot γ, so γ isotopic to pt × S1 ⊂ S3 × S1. ◮ Hence, can cancel 1-h and 2-h, so DW = ∂(0-h) = S4. 8
SLIDE 65
Mazur manifolds
Theorem (Akbulut-Kirby, 1978)
Σ(3, 4, 5) embeds in S4.
4
Proof:
◮ Σ(3, 4, 5) is the boundary of W 4 pictured ◮ W 4 is a Mazur manifold, i.e. a contractible 4-mfld built from a 0-h, a
1-h and a 2-h.
◮ Claim: The double DW of a Mazur manifold W is S4, hence W
embeds in S4. Proof of claim: (Mazur, 1960)
◮ DW = W ∪∂ (−W ) = ∂(W × [0, 1]) ◮ W × [0, 1] is a 5-manifold built from a 0-h, 1-h, 2-h. ◮ 2-h attached along knot γ in ∂(0-h ∪ 1-h) = S3 × S1. ◮ Can unknot γ, so γ isotopic to pt × S1 ⊂ S3 × S1. ◮ Hence, can cancel 1-h and 2-h, so DW = ∂(0-h) = S4. 8
SLIDE 66
Mazur manifolds
Theorem (Akbulut-Kirby, 1978)
Σ(3, 4, 5) embeds in S4.
4
Proof:
◮ Σ(3, 4, 5) is the boundary of W 4 pictured ◮ W 4 is a Mazur manifold, i.e. a contractible 4-mfld built from a 0-h, a
1-h and a 2-h.
◮ Claim: The double DW of a Mazur manifold W is S4, hence W
embeds in S4. Proof of claim: (Mazur, 1960)
◮ DW = W ∪∂ (−W ) = ∂(W × [0, 1]) ◮ W × [0, 1] is a 5-manifold built from a 0-h, 1-h, 2-h. ◮ 2-h attached along knot γ in ∂(0-h ∪ 1-h) = S3 × S1. ◮ Can unknot γ, so γ isotopic to pt × S1 ⊂ S3 × S1. ◮ Hence, can cancel 1-h and 2-h, so DW = ∂(0-h) = S4. 8
SLIDE 67
Mazur manifolds
Theorem (Akbulut-Kirby, 1978)
Σ(3, 4, 5) embeds in S4.
4
Proof:
◮ Σ(3, 4, 5) is the boundary of W 4 pictured ◮ W 4 is a Mazur manifold, i.e. a contractible 4-mfld built from a 0-h, a
1-h and a 2-h.
◮ Claim: The double DW of a Mazur manifold W is S4, hence W
embeds in S4. Proof of claim: (Mazur, 1960)
◮ DW = W ∪∂ (−W ) = ∂(W × [0, 1]) ◮ W × [0, 1] is a 5-manifold built from a 0-h, 1-h, 2-h. ◮ 2-h attached along knot γ in ∂(0-h ∪ 1-h) = S3 × S1. ◮ Can unknot γ, so γ isotopic to pt × S1 ⊂ S3 × S1. ◮ Hence, can cancel 1-h and 2-h, so DW = ∂(0-h) = S4. 8
SLIDE 68
Mazur manifolds
Theorem (Akbulut-Kirby, 1978)
Σ(3, 4, 5) embeds in S4.
4
Proof:
◮ Σ(3, 4, 5) is the boundary of W 4 pictured ◮ W 4 is a Mazur manifold, i.e. a contractible 4-mfld built from a 0-h, a
1-h and a 2-h.
◮ Claim: The double DW of a Mazur manifold W is S4, hence W
embeds in S4. Proof of claim: (Mazur, 1960)
◮ DW = W ∪∂ (−W ) = ∂(W × [0, 1]) ◮ W × [0, 1] is a 5-manifold built from a 0-h, 1-h, 2-h. ◮ 2-h attached along knot γ in ∂(0-h ∪ 1-h) = S3 × S1. ◮ Can unknot γ, so γ isotopic to pt × S1 ⊂ S3 × S1. ◮ Hence, can cancel 1-h and 2-h, so DW = ∂(0-h) = S4. 8
SLIDE 69
Theorem (Casson-Harer, 1978)
Each of the following Brieskorn spheres bound Mazur manifolds:
◮ Σ(p, ps − 1, ps + 1), p even, s odd, and ◮ Σ(p, ps ± 1, ps ± 2), p odd, s arbitrary.
This family includes for example Σ(3, 4, 5), Σ(3, 7, 8), Σ(5, 6, 7).
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Example: Σ(3, 4, 5) is the double branched cover of the Montesinos knot:
9
SLIDE 70
Theorem (Casson-Harer, 1978)
Each of the following Brieskorn spheres bound Mazur manifolds:
◮ Σ(p, ps − 1, ps + 1), p even, s odd, and ◮ Σ(p, ps ± 1, ps ± 2), p odd, s arbitrary.
This family includes for example Σ(3, 4, 5), Σ(3, 7, 8), Σ(5, 6, 7).
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Example: Σ(3, 4, 5) is the double branched cover of the Montesinos knot:
9
SLIDE 71
Theorem (Casson-Harer, 1978)
Each of the following Brieskorn spheres bound Mazur manifolds:
◮ Σ(p, ps − 1, ps + 1), p even, s odd, and ◮ Σ(p, ps ± 1, ps ± 2), p odd, s arbitrary.
This family includes for example Σ(3, 4, 5), Σ(3, 7, 8), Σ(5, 6, 7).
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Example: Σ(3, 4, 5) is the double branched cover of the Montesinos knot:
9
SLIDE 72
Theorem (Casson-Harer, 1978)
Each of the following Brieskorn spheres bound Mazur manifolds:
◮ Σ(p, ps − 1, ps + 1), p even, s odd, and ◮ Σ(p, ps ± 1, ps ± 2), p odd, s arbitrary.
This family includes for example Σ(3, 4, 5), Σ(3, 7, 8), Σ(5, 6, 7).
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Example: Σ(3, 4, 5) is the double branched cover of the Montesinos knot:
9
SLIDE 73
Theorem (Casson-Harer, 1978)
Each of the following Brieskorn spheres bound Mazur manifolds:
◮ Σ(p, ps − 1, ps + 1), p even, s odd, and ◮ Σ(p, ps ± 1, ps ± 2), p odd, s arbitrary.
This family includes for example Σ(3, 4, 5), Σ(3, 7, 8), Σ(5, 6, 7).
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Example: Σ(3, 4, 5) is the double branched cover of the Montesinos knot:
9
SLIDE 74
Theorem (Casson-Harer, 1978)
Each of the following Brieskorn spheres bound Mazur manifolds:
◮ Σ(p, ps − 1, ps + 1), p even, s odd, and ◮ Σ(p, ps ± 1, ps ± 2), p odd, s arbitrary.
This family includes for example Σ(3, 4, 5), Σ(3, 7, 8), Σ(5, 6, 7).
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Example: Σ(3, 4, 5) is the double branched cover of the Montesinos knot:
9
SLIDE 75
Theorem (Casson-Harer, 1978)
Each of the following Brieskorn spheres bound Mazur manifolds:
◮ Σ(p, ps − 1, ps + 1), p even, s odd, and ◮ Σ(p, ps ± 1, ps ± 2), p odd, s arbitrary.
This family includes for example Σ(3, 4, 5), Σ(3, 7, 8), Σ(5, 6, 7).
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Example: Σ(3, 4, 5) is the double branched cover of the Montesinos knot:
9
SLIDE 76
I’ll sketch a proof of Casson-Harer’s theorem.
Lemma 1
Proof:
10
SLIDE 77
I’ll sketch a proof of Casson-Harer’s theorem.
Lemma 1
branched double cover
Proof:
10
SLIDE 78
I’ll sketch a proof of Casson-Harer’s theorem.
Lemma 1
branched double cover
Proof:
π rotation
10
SLIDE 79
I’ll sketch a proof of Casson-Harer’s theorem.
Lemma 1
branched double cover
Proof:
π rotation fold fold
10
SLIDE 80
I’ll sketch a proof of Casson-Harer’s theorem.
Lemma 1
branched double cover
Proof:
π rotation fold fold
=
10
SLIDE 81
Lemma 2
S1 × S2 is the double branched cover of the 2 component unlink in S3. Proof:
◮ S3 = B1 ∪ B2 ◮ Passing to double branched covers: Σ2(S3) = Σ2(B1) ∪ Σ2(B2). ◮ Σ2(S3) is two solid tori glued along boundaries by identity map. ◮ Σ2(S3) = S1 × S2. 11
SLIDE 82
Lemma 2
S1 × S2 is the double branched cover of the 2 component unlink in S3. Proof:
◮ S3 = B1 ∪ B2 ◮ Passing to double branched covers: Σ2(S3) = Σ2(B1) ∪ Σ2(B2). ◮ Σ2(S3) is two solid tori glued along boundaries by identity map. ◮ Σ2(S3) = S1 × S2. 11
SLIDE 83
Lemma 2
S1 × S2 is the double branched cover of the 2 component unlink in S3. Proof:
◮ S3 = B1 ∪ B2 ◮ Passing to double branched covers: Σ2(S3) = Σ2(B1) ∪ Σ2(B2). ◮ Σ2(S3) is two solid tori glued along boundaries by identity map. ◮ Σ2(S3) = S1 × S2.
B1 B2
11
SLIDE 84
Lemma 2
S1 × S2 is the double branched cover of the 2 component unlink in S3. Proof:
◮ S3 = B1 ∪ B2 ◮ Passing to double branched covers: Σ2(S3) = Σ2(B1) ∪ Σ2(B2). ◮ Σ2(S3) is two solid tori glued along boundaries by identity map. ◮ Σ2(S3) = S1 × S2.
B1 B2
11
SLIDE 85
Lemma 2
S1 × S2 is the double branched cover of the 2 component unlink in S3. Proof:
◮ S3 = B1 ∪ B2 ◮ Passing to double branched covers: Σ2(S3) = Σ2(B1) ∪ Σ2(B2). ◮ Σ2(S3) is two solid tori glued along boundaries by identity map. ◮ Σ2(S3) = S1 × S2.
B1 B2
glue by identity
S1 x S2 branched double cover
11
SLIDE 86
Lemma 2
S1 × S2 is the double branched cover of the 2 component unlink in S3. Proof:
◮ S3 = B1 ∪ B2 ◮ Passing to double branched covers: Σ2(S3) = Σ2(B1) ∪ Σ2(B2). ◮ Σ2(S3) is two solid tori glued along boundaries by identity map. ◮ Σ2(S3) = S1 × S2.
B1 B2
glue by identity
S1 x S2 branched double cover
11
SLIDE 87
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Proof:
◮ Claim: M3 is surgery on a knot in S1 × S2. ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K. ◮ Downstairs: Replace (B3, 2-arcs) with another (B3, 2-arcs). ◮ Upstairs: Replace solid torus in S1 × S2 with another solid torus.
Change 0-framed unknot to dotted circle to get Mazur manifold.
12
SLIDE 88
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Proof:
◮ Claim: M3 is surgery on a knot in S1 × S2. ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K. ◮ Downstairs: Replace (B3, 2-arcs) with another (B3, 2-arcs). ◮ Upstairs: Replace solid torus in S1 × S2 with another solid torus.
Change 0-framed unknot to dotted circle to get Mazur manifold.
12
SLIDE 89
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Proof:
◮ Claim: M3 is surgery on a knot in S1 × S2. ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K. ◮ Downstairs: Replace (B3, 2-arcs) with another (B3, 2-arcs). ◮ Upstairs: Replace solid torus in S1 × S2 with another solid torus.
Change 0-framed unknot to dotted circle to get Mazur manifold.
12
SLIDE 90
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Proof:
◮ Claim: M3 is surgery on a knot in S1 × S2. ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K. ◮ Downstairs: Replace (B3, 2-arcs) with another (B3, 2-arcs). ◮ Upstairs: Replace solid torus in S1 × S2 with another solid torus.
Change 0-framed unknot to dotted circle to get Mazur manifold.
12
SLIDE 91
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Proof:
◮ Claim: M3 is surgery on a knot in S1 × S2. ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K. ◮ Downstairs: Replace (B3, 2-arcs) with another (B3, 2-arcs). ◮ Upstairs: Replace solid torus in S1 × S2 with another solid torus.
Change 0-framed unknot to dotted circle to get Mazur manifold.
12
SLIDE 92
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Proof:
◮ Claim: M3 is surgery on a knot in S1 × S2. ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K. ◮ Downstairs: Replace (B3, 2-arcs) with another (B3, 2-arcs). ◮ Upstairs: Replace solid torus in S1 × S2 with another solid torus.
Change 0-framed unknot to dotted circle to get Mazur manifold.
canceling band move
12
SLIDE 93
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Proof:
◮ Claim: M3 is surgery on a knot in S1 × S2. ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K. ◮ Downstairs: Replace (B3, 2-arcs) with another (B3, 2-arcs). ◮ Upstairs: Replace solid torus in S1 × S2 with another solid torus.
Change 0-framed unknot to dotted circle to get Mazur manifold.
12
SLIDE 94
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Proof:
◮ Claim: M3 is surgery on a knot in S1 × S2. ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K. ◮ Downstairs: Replace (B3, 2-arcs) with another (B3, 2-arcs). ◮ Upstairs: Replace solid torus in S1 × S2 with another solid torus.
Change 0-framed unknot to dotted circle to get Mazur manifold.
B3 12
SLIDE 95
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Proof:
◮ Claim: M3 is surgery on a knot in S1 × S2. ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K. ◮ Downstairs: Replace (B3, 2-arcs) with another (B3, 2-arcs). ◮ Upstairs: Replace solid torus in S1 × S2 with another solid torus.
Change 0-framed unknot to dotted circle to get Mazur manifold.
B3
remove this ball (torus in double branched cover)
12
SLIDE 96
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Proof:
◮ Claim: M3 is surgery on a knot in S1 × S2. ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K. ◮ Downstairs: Replace (B3, 2-arcs) with another (B3, 2-arcs). ◮ Upstairs: Replace solid torus in S1 × S2 with another solid torus.
Change 0-framed unknot to dotted circle to get Mazur manifold.
B3
glue in this ball (torus in double branched cover)
12
SLIDE 97
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Proof:
◮ Claim: M3 is surgery on a knot in S1 × S2. ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K. ◮ Downstairs: Replace (B3, 2-arcs) with another (B3, 2-arcs). ◮ Upstairs: Replace solid torus in S1 × S2 with another solid torus.
Change 0-framed unknot to dotted circle to get Mazur manifold.
4
B3
glue in this ball (torus in double branched cover)
12
SLIDE 98
Theorem (Casson-Harer)
Let M3 be a ZHS which is the double branched cover of a knot K in S3. If K is ribbon via a single band move then M3 is the boundary of a Mazur manifold. Proof:
◮ Claim: M3 is surgery on a knot in S1 × S2. ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K. ◮ Downstairs: Replace (B3, 2-arcs) with another (B3, 2-arcs). ◮ Upstairs: Replace solid torus in S1 × S2 with another solid torus.
Change 0-framed unknot to dotted circle to get Mazur manifold.
4
B3
glue in this ball (torus in double branched cover)
12
SLIDE 99
All Σ(p, q, r) known to bound a ZHB4 belong to one of:
◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ∓ 1), p even, s odd, k ≥ 0. ◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ± 2), p odd, s arbitrary, k ≥ 0. ◮ Σ(p, ps ± 2, k · p(ps ± 2) + ps ± 1), p odd, s arbitrary, k ≥ 0.
Casson-Harer implies k = 0 all bound Mazur manifolds.
Theorem (Stern, 1978)
The subfamilies with k = 2 and p = 2, 3 bound contractible manifolds built with a 0-h, two 1-h’s, two 2-h’s.
Theorem (Fickle, 1982)
k = 2 and p = 2, 3 (i.e. Stern’s examples) bound Mazur manifolds. Apart from these, there are five other Σ(p, q, r) known to bound a ZHB4:
◮ (Fintushel-Stern) Σ(2, 7, 19), Σ(3, 5, 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2, 7, 47), Σ(3, 5, 49) bound ZHB4’s. ◮ (Fickle) Σ(2, 3, 25) bounds a Mazur manifold.
Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2, 3 and k even bound ZHB4’s.
13
SLIDE 100
All Σ(p, q, r) known to bound a ZHB4 belong to one of:
◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ∓ 1), p even, s odd, k ≥ 0. ◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ± 2), p odd, s arbitrary, k ≥ 0. ◮ Σ(p, ps ± 2, k · p(ps ± 2) + ps ± 1), p odd, s arbitrary, k ≥ 0.
Casson-Harer implies k = 0 all bound Mazur manifolds.
Theorem (Stern, 1978)
The subfamilies with k = 2 and p = 2, 3 bound contractible manifolds built with a 0-h, two 1-h’s, two 2-h’s.
Theorem (Fickle, 1982)
k = 2 and p = 2, 3 (i.e. Stern’s examples) bound Mazur manifolds. Apart from these, there are five other Σ(p, q, r) known to bound a ZHB4:
◮ (Fintushel-Stern) Σ(2, 7, 19), Σ(3, 5, 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2, 7, 47), Σ(3, 5, 49) bound ZHB4’s. ◮ (Fickle) Σ(2, 3, 25) bounds a Mazur manifold.
Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2, 3 and k even bound ZHB4’s.
13
SLIDE 101
All Σ(p, q, r) known to bound a ZHB4 belong to one of:
◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ∓ 1), p even, s odd, k ≥ 0. ◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ± 2), p odd, s arbitrary, k ≥ 0. ◮ Σ(p, ps ± 2, k · p(ps ± 2) + ps ± 1), p odd, s arbitrary, k ≥ 0.
Casson-Harer implies k = 0 all bound Mazur manifolds.
Theorem (Stern, 1978)
The subfamilies with k = 2 and p = 2, 3 bound contractible manifolds built with a 0-h, two 1-h’s, two 2-h’s.
Theorem (Fickle, 1982)
k = 2 and p = 2, 3 (i.e. Stern’s examples) bound Mazur manifolds. Apart from these, there are five other Σ(p, q, r) known to bound a ZHB4:
◮ (Fintushel-Stern) Σ(2, 7, 19), Σ(3, 5, 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2, 7, 47), Σ(3, 5, 49) bound ZHB4’s. ◮ (Fickle) Σ(2, 3, 25) bounds a Mazur manifold.
Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2, 3 and k even bound ZHB4’s.
13
SLIDE 102
All Σ(p, q, r) known to bound a ZHB4 belong to one of:
◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ∓ 1), p even, s odd, k ≥ 0. ◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ± 2), p odd, s arbitrary, k ≥ 0. ◮ Σ(p, ps ± 2, k · p(ps ± 2) + ps ± 1), p odd, s arbitrary, k ≥ 0.
Casson-Harer implies k = 0 all bound Mazur manifolds.
Theorem (Stern, 1978)
The subfamilies with k = 2 and p = 2, 3 bound contractible manifolds built with a 0-h, two 1-h’s, two 2-h’s.
Theorem (Fickle, 1982)
k = 2 and p = 2, 3 (i.e. Stern’s examples) bound Mazur manifolds. Apart from these, there are five other Σ(p, q, r) known to bound a ZHB4:
◮ (Fintushel-Stern) Σ(2, 7, 19), Σ(3, 5, 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2, 7, 47), Σ(3, 5, 49) bound ZHB4’s. ◮ (Fickle) Σ(2, 3, 25) bounds a Mazur manifold.
Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2, 3 and k even bound ZHB4’s.
13
SLIDE 103
All Σ(p, q, r) known to bound a ZHB4 belong to one of:
◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ∓ 1), p even, s odd, k ≥ 0. ◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ± 2), p odd, s arbitrary, k ≥ 0. ◮ Σ(p, ps ± 2, k · p(ps ± 2) + ps ± 1), p odd, s arbitrary, k ≥ 0.
Casson-Harer implies k = 0 all bound Mazur manifolds.
Theorem (Stern, 1978)
The subfamilies with k = 2 and p = 2, 3 bound contractible manifolds built with a 0-h, two 1-h’s, two 2-h’s.
Theorem (Fickle, 1982)
k = 2 and p = 2, 3 (i.e. Stern’s examples) bound Mazur manifolds. Apart from these, there are five other Σ(p, q, r) known to bound a ZHB4:
◮ (Fintushel-Stern) Σ(2, 7, 19), Σ(3, 5, 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2, 7, 47), Σ(3, 5, 49) bound ZHB4’s. ◮ (Fickle) Σ(2, 3, 25) bounds a Mazur manifold.
Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2, 3 and k even bound ZHB4’s.
13
SLIDE 104
All Σ(p, q, r) known to bound a ZHB4 belong to one of:
◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ∓ 1), p even, s odd, k ≥ 0. ◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ± 2), p odd, s arbitrary, k ≥ 0. ◮ Σ(p, ps ± 2, k · p(ps ± 2) + ps ± 1), p odd, s arbitrary, k ≥ 0.
Casson-Harer implies k = 0 all bound Mazur manifolds.
Theorem (Stern, 1978)
The subfamilies with k = 2 and p = 2, 3 bound contractible manifolds built with a 0-h, two 1-h’s, two 2-h’s.
Theorem (Fickle, 1982)
k = 2 and p = 2, 3 (i.e. Stern’s examples) bound Mazur manifolds. Apart from these, there are five other Σ(p, q, r) known to bound a ZHB4:
◮ (Fintushel-Stern) Σ(2, 7, 19), Σ(3, 5, 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2, 7, 47), Σ(3, 5, 49) bound ZHB4’s. ◮ (Fickle) Σ(2, 3, 25) bounds a Mazur manifold.
Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2, 3 and k even bound ZHB4’s.
13
SLIDE 105
All Σ(p, q, r) known to bound a ZHB4 belong to one of:
◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ∓ 1), p even, s odd, k ≥ 0. ◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ± 2), p odd, s arbitrary, k ≥ 0. ◮ Σ(p, ps ± 2, k · p(ps ± 2) + ps ± 1), p odd, s arbitrary, k ≥ 0.
Casson-Harer implies k = 0 all bound Mazur manifolds.
Theorem (Stern, 1978)
The subfamilies with k = 2 and p = 2, 3 bound contractible manifolds built with a 0-h, two 1-h’s, two 2-h’s.
Theorem (Fickle, 1982)
k = 2 and p = 2, 3 (i.e. Stern’s examples) bound Mazur manifolds. Apart from these, there are five other Σ(p, q, r) known to bound a ZHB4:
◮ (Fintushel-Stern) Σ(2, 7, 19), Σ(3, 5, 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2, 7, 47), Σ(3, 5, 49) bound ZHB4’s. ◮ (Fickle) Σ(2, 3, 25) bounds a Mazur manifold.
Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2, 3 and k even bound ZHB4’s.
13
SLIDE 106
All Σ(p, q, r) known to bound a ZHB4 belong to one of:
◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ∓ 1), p even, s odd, k ≥ 0. ◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ± 2), p odd, s arbitrary, k ≥ 0. ◮ Σ(p, ps ± 2, k · p(ps ± 2) + ps ± 1), p odd, s arbitrary, k ≥ 0.
Casson-Harer implies k = 0 all bound Mazur manifolds.
Theorem (Stern, 1978)
The subfamilies with k = 2 and p = 2, 3 bound contractible manifolds built with a 0-h, two 1-h’s, two 2-h’s.
Theorem (Fickle, 1982)
k = 2 and p = 2, 3 (i.e. Stern’s examples) bound Mazur manifolds. Apart from these, there are five other Σ(p, q, r) known to bound a ZHB4:
◮ (Fintushel-Stern) Σ(2, 7, 19), Σ(3, 5, 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2, 7, 47), Σ(3, 5, 49) bound ZHB4’s. ◮ (Fickle) Σ(2, 3, 25) bounds a Mazur manifold.
Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2, 3 and k even bound ZHB4’s.
13
SLIDE 107
All Σ(p, q, r) known to bound a ZHB4 belong to one of:
◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ∓ 1), p even, s odd, k ≥ 0. ◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ± 2), p odd, s arbitrary, k ≥ 0. ◮ Σ(p, ps ± 2, k · p(ps ± 2) + ps ± 1), p odd, s arbitrary, k ≥ 0.
Casson-Harer implies k = 0 all bound Mazur manifolds.
Theorem (Stern, 1978)
The subfamilies with k = 2 and p = 2, 3 bound contractible manifolds built with a 0-h, two 1-h’s, two 2-h’s.
Theorem (Fickle, 1982)
k = 2 and p = 2, 3 (i.e. Stern’s examples) bound Mazur manifolds. Apart from these, there are five other Σ(p, q, r) known to bound a ZHB4:
◮ (Fintushel-Stern) Σ(2, 7, 19), Σ(3, 5, 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2, 7, 47), Σ(3, 5, 49) bound ZHB4’s. ◮ (Fickle) Σ(2, 3, 25) bounds a Mazur manifold.
Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2, 3 and k even bound ZHB4’s.
13
SLIDE 108
All Σ(p, q, r) known to bound a ZHB4 belong to one of:
◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ∓ 1), p even, s odd, k ≥ 0. ◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ± 2), p odd, s arbitrary, k ≥ 0. ◮ Σ(p, ps ± 2, k · p(ps ± 2) + ps ± 1), p odd, s arbitrary, k ≥ 0.
Casson-Harer implies k = 0 all bound Mazur manifolds.
Theorem (Stern, 1978)
The subfamilies with k = 2 and p = 2, 3 bound contractible manifolds built with a 0-h, two 1-h’s, two 2-h’s.
Theorem (Fickle, 1982)
k = 2 and p = 2, 3 (i.e. Stern’s examples) bound Mazur manifolds. Apart from these, there are five other Σ(p, q, r) known to bound a ZHB4:
◮ (Fintushel-Stern) Σ(2, 7, 19), Σ(3, 5, 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2, 7, 47), Σ(3, 5, 49) bound ZHB4’s. ◮ (Fickle) Σ(2, 3, 25) bounds a Mazur manifold.
Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2, 3 and k even bound ZHB4’s.
13
SLIDE 109
All Σ(p, q, r) known to bound a ZHB4 belong to one of:
◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ∓ 1), p even, s odd, k ≥ 0. ◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ± 2), p odd, s arbitrary, k ≥ 0. ◮ Σ(p, ps ± 2, k · p(ps ± 2) + ps ± 1), p odd, s arbitrary, k ≥ 0.
Casson-Harer implies k = 0 all bound Mazur manifolds.
Theorem (Stern, 1978)
The subfamilies with k = 2 and p = 2, 3 bound contractible manifolds built with a 0-h, two 1-h’s, two 2-h’s.
Theorem (Fickle, 1982)
k = 2 and p = 2, 3 (i.e. Stern’s examples) bound Mazur manifolds. Apart from these, there are five other Σ(p, q, r) known to bound a ZHB4:
◮ (Fintushel-Stern) Σ(2, 7, 19), Σ(3, 5, 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2, 7, 47), Σ(3, 5, 49) bound ZHB4’s. ◮ (Fickle) Σ(2, 3, 25) bounds a Mazur manifold.
Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2, 3 and k even bound ZHB4’s.
13
SLIDE 110
All Σ(p, q, r) known to bound a ZHB4 belong to one of:
◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ∓ 1), p even, s odd, k ≥ 0. ◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ± 2), p odd, s arbitrary, k ≥ 0. ◮ Σ(p, ps ± 2, k · p(ps ± 2) + ps ± 1), p odd, s arbitrary, k ≥ 0.
Casson-Harer implies k = 0 all bound Mazur manifolds.
Theorem (Stern, 1978)
The subfamilies with k = 2 and p = 2, 3 bound contractible manifolds built with a 0-h, two 1-h’s, two 2-h’s.
Theorem (Fickle, 1982)
k = 2 and p = 2, 3 (i.e. Stern’s examples) bound Mazur manifolds. Apart from these, there are five other Σ(p, q, r) known to bound a ZHB4:
◮ (Fintushel-Stern) Σ(2, 7, 19), Σ(3, 5, 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2, 7, 47), Σ(3, 5, 49) bound ZHB4’s. ◮ (Fickle) Σ(2, 3, 25) bounds a Mazur manifold.
Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2, 3 and k even bound ZHB4’s.
13
SLIDE 111
All Σ(p, q, r) known to bound a ZHB4 belong to one of:
◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ∓ 1), p even, s odd, k ≥ 0. ◮ Σ(p, ps ± 1, k · p(ps ± 1) + ps ± 2), p odd, s arbitrary, k ≥ 0. ◮ Σ(p, ps ± 2, k · p(ps ± 2) + ps ± 1), p odd, s arbitrary, k ≥ 0.
Casson-Harer implies k = 0 all bound Mazur manifolds.
Theorem (Stern, 1978)
The subfamilies with k = 2 and p = 2, 3 bound contractible manifolds built with a 0-h, two 1-h’s, two 2-h’s.
Theorem (Fickle, 1982)
k = 2 and p = 2, 3 (i.e. Stern’s examples) bound Mazur manifolds. Apart from these, there are five other Σ(p, q, r) known to bound a ZHB4:
◮ (Fintushel-Stern) Σ(2, 7, 19), Σ(3, 5, 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2, 7, 47), Σ(3, 5, 49) bound ZHB4’s. ◮ (Fickle) Σ(2, 3, 25) bounds a Mazur manifold.
Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2, 3 and k even bound ZHB4’s.
13
SLIDE 112
Fickle proved all of Stern’s examples bound Mazur manifolds using the following:
Theorem (Fickle)
◮ Suppose K ⊂ S3 has genus one Seifert surface F. ◮ Suppose b ⊂ F is an essential simple closed curve, unknotted in S3.
Then, if b has self-linking s, then
1 s±1 surgery on K bounds a Mazur
manifold. Example: Σ(2, 3, 13) is −1/2 surgery on the right handed trefoil:
14
SLIDE 113
Fickle proved all of Stern’s examples bound Mazur manifolds using the following:
Theorem (Fickle)
◮ Suppose K ⊂ S3 has genus one Seifert surface F. ◮ Suppose b ⊂ F is an essential simple closed curve, unknotted in S3.
Then, if b has self-linking s, then
1 s±1 surgery on K bounds a Mazur
manifold. Example: Σ(2, 3, 13) is −1/2 surgery on the right handed trefoil:
14
SLIDE 114
Fickle proved all of Stern’s examples bound Mazur manifolds using the following:
Theorem (Fickle)
◮ Suppose K ⊂ S3 has genus one Seifert surface F. ◮ Suppose b ⊂ F is an essential simple closed curve, unknotted in S3.
Then, if b has self-linking s, then
1 s±1 surgery on K bounds a Mazur
manifold. Example: Σ(2, 3, 13) is −1/2 surgery on the right handed trefoil:
14
SLIDE 115
Fickle proved all of Stern’s examples bound Mazur manifolds using the following:
Theorem (Fickle)
◮ Suppose K ⊂ S3 has genus one Seifert surface F. ◮ Suppose b ⊂ F is an essential simple closed curve, unknotted in S3.
Then, if b has self-linking s, then
1 s±1 surgery on K bounds a Mazur
manifold. Example: Σ(2, 3, 13) is −1/2 surgery on the right handed trefoil:
14
SLIDE 116
Fickle proved all of Stern’s examples bound Mazur manifolds using the following:
Theorem (Fickle)
◮ Suppose K ⊂ S3 has genus one Seifert surface F. ◮ Suppose b ⊂ F is an essential simple closed curve, unknotted in S3.
Then, if b has self-linking s, then
1 s±1 surgery on K bounds a Mazur
manifold. Example: Σ(2, 3, 13) is −1/2 surgery on the right handed trefoil:
14
SLIDE 117
Fickle proved all of Stern’s examples bound Mazur manifolds using the following:
Theorem (Fickle)
◮ Suppose K ⊂ S3 has genus one Seifert surface F. ◮ Suppose b ⊂ F is an essential simple closed curve, unknotted in S3.
Then, if b has self-linking s, then
1 s±1 surgery on K bounds a Mazur
manifold. Example: Σ(2, 3, 13) is −1/2 surgery on the right handed trefoil:
K = RH trefoil
14
SLIDE 118
Fickle proved all of Stern’s examples bound Mazur manifolds using the following:
Theorem (Fickle)
◮ Suppose K ⊂ S3 has genus one Seifert surface F. ◮ Suppose b ⊂ F is an essential simple closed curve, unknotted in S3.
Then, if b has self-linking s, then
1 s±1 surgery on K bounds a Mazur
manifold. Example: Σ(2, 3, 13) is −1/2 surgery on the right handed trefoil:
K = RH trefoil b self-linking = -1
14
SLIDE 119
Questions and final remarks
Q: What about embedding Σ(a1, . . . , an), n ≥ 4?
◮ Conjecture: (Kollar) None bound acyclic manifolds, in particular
none embed in S4.
◮ Related to the Montgomery-Yang conjecture.
Q: Which Montesinos knots with homology sphere double branched cover are slice?
◮ Casson-Harer’s list is incomplete: I found that the Montesinos knot
corresponding to Σ(2, 7, 19) is ribbon.
◮ For 4 or more parameters, if the above conjecture is true then none
are ribbon. Q: Are there further obstructions to a ZHS bounding a Mazur manifold (equiv. being surgery on a knot in S1 × S2)?
◮ A corollary of Taubes theorem on end-periodic manifolds is that
Σ(2, 3, 5)#Σ(2, 3, 5) cannot bound a contractible manifold.
◮ However, this manifold embeds in S4.
Q: Give an example of a ZHS3 which embeds in a ZHS4 but not in S4. Q: Does Σ(3, 5, 8) embed in S4?
15
SLIDE 120
Questions and final remarks
Q: What about embedding Σ(a1, . . . , an), n ≥ 4?
◮ Conjecture: (Kollar) None bound acyclic manifolds, in particular
none embed in S4.
◮ Related to the Montgomery-Yang conjecture.
Q: Which Montesinos knots with homology sphere double branched cover are slice?
◮ Casson-Harer’s list is incomplete: I found that the Montesinos knot
corresponding to Σ(2, 7, 19) is ribbon.
◮ For 4 or more parameters, if the above conjecture is true then none
are ribbon. Q: Are there further obstructions to a ZHS bounding a Mazur manifold (equiv. being surgery on a knot in S1 × S2)?
◮ A corollary of Taubes theorem on end-periodic manifolds is that
Σ(2, 3, 5)#Σ(2, 3, 5) cannot bound a contractible manifold.
◮ However, this manifold embeds in S4.
Q: Give an example of a ZHS3 which embeds in a ZHS4 but not in S4. Q: Does Σ(3, 5, 8) embed in S4?
15
SLIDE 121
Questions and final remarks
Q: What about embedding Σ(a1, . . . , an), n ≥ 4?
◮ Conjecture: (Kollar) None bound acyclic manifolds, in particular
none embed in S4.
◮ Related to the Montgomery-Yang conjecture.
Q: Which Montesinos knots with homology sphere double branched cover are slice?
◮ Casson-Harer’s list is incomplete: I found that the Montesinos knot
corresponding to Σ(2, 7, 19) is ribbon.
◮ For 4 or more parameters, if the above conjecture is true then none
are ribbon. Q: Are there further obstructions to a ZHS bounding a Mazur manifold (equiv. being surgery on a knot in S1 × S2)?
◮ A corollary of Taubes theorem on end-periodic manifolds is that
Σ(2, 3, 5)#Σ(2, 3, 5) cannot bound a contractible manifold.
◮ However, this manifold embeds in S4.
Q: Give an example of a ZHS3 which embeds in a ZHS4 but not in S4. Q: Does Σ(3, 5, 8) embed in S4?
15
SLIDE 122
Questions and final remarks
Q: What about embedding Σ(a1, . . . , an), n ≥ 4?
◮ Conjecture: (Kollar) None bound acyclic manifolds, in particular
none embed in S4.
◮ Related to the Montgomery-Yang conjecture.
Q: Which Montesinos knots with homology sphere double branched cover are slice?
◮ Casson-Harer’s list is incomplete: I found that the Montesinos knot
corresponding to Σ(2, 7, 19) is ribbon.
◮ For 4 or more parameters, if the above conjecture is true then none
are ribbon. Q: Are there further obstructions to a ZHS bounding a Mazur manifold (equiv. being surgery on a knot in S1 × S2)?
◮ A corollary of Taubes theorem on end-periodic manifolds is that
Σ(2, 3, 5)#Σ(2, 3, 5) cannot bound a contractible manifold.
◮ However, this manifold embeds in S4.
Q: Give an example of a ZHS3 which embeds in a ZHS4 but not in S4. Q: Does Σ(3, 5, 8) embed in S4?
15
SLIDE 123
Questions and final remarks
Q: What about embedding Σ(a1, . . . , an), n ≥ 4?
◮ Conjecture: (Kollar) None bound acyclic manifolds, in particular
none embed in S4.
◮ Related to the Montgomery-Yang conjecture.
Q: Which Montesinos knots with homology sphere double branched cover are slice?
◮ Casson-Harer’s list is incomplete: I found that the Montesinos knot
corresponding to Σ(2, 7, 19) is ribbon.
◮ For 4 or more parameters, if the above conjecture is true then none
are ribbon. Q: Are there further obstructions to a ZHS bounding a Mazur manifold (equiv. being surgery on a knot in S1 × S2)?
◮ A corollary of Taubes theorem on end-periodic manifolds is that
Σ(2, 3, 5)#Σ(2, 3, 5) cannot bound a contractible manifold.
◮ However, this manifold embeds in S4.
Q: Give an example of a ZHS3 which embeds in a ZHS4 but not in S4. Q: Does Σ(3, 5, 8) embed in S4?
15
SLIDE 124
Questions and final remarks
Q: What about embedding Σ(a1, . . . , an), n ≥ 4?
◮ Conjecture: (Kollar) None bound acyclic manifolds, in particular
none embed in S4.
◮ Related to the Montgomery-Yang conjecture.
Q: Which Montesinos knots with homology sphere double branched cover are slice?
◮ Casson-Harer’s list is incomplete: I found that the Montesinos knot
corresponding to Σ(2, 7, 19) is ribbon.
◮ For 4 or more parameters, if the above conjecture is true then none
are ribbon. Q: Are there further obstructions to a ZHS bounding a Mazur manifold (equiv. being surgery on a knot in S1 × S2)?
◮ A corollary of Taubes theorem on end-periodic manifolds is that
Σ(2, 3, 5)#Σ(2, 3, 5) cannot bound a contractible manifold.
◮ However, this manifold embeds in S4.
Q: Give an example of a ZHS3 which embeds in a ZHS4 but not in S4. Q: Does Σ(3, 5, 8) embed in S4?
15
SLIDE 125
Questions and final remarks
Q: What about embedding Σ(a1, . . . , an), n ≥ 4?
◮ Conjecture: (Kollar) None bound acyclic manifolds, in particular
none embed in S4.
◮ Related to the Montgomery-Yang conjecture.
Q: Which Montesinos knots with homology sphere double branched cover are slice?
◮ Casson-Harer’s list is incomplete: I found that the Montesinos knot
corresponding to Σ(2, 7, 19) is ribbon.
◮ For 4 or more parameters, if the above conjecture is true then none
are ribbon. Q: Are there further obstructions to a ZHS bounding a Mazur manifold (equiv. being surgery on a knot in S1 × S2)?
◮ A corollary of Taubes theorem on end-periodic manifolds is that
Σ(2, 3, 5)#Σ(2, 3, 5) cannot bound a contractible manifold.
◮ However, this manifold embeds in S4.
Q: Give an example of a ZHS3 which embeds in a ZHS4 but not in S4. Q: Does Σ(3, 5, 8) embed in S4?
15
SLIDE 126
Questions and final remarks
Q: What about embedding Σ(a1, . . . , an), n ≥ 4?
◮ Conjecture: (Kollar) None bound acyclic manifolds, in particular
none embed in S4.
◮ Related to the Montgomery-Yang conjecture.
Q: Which Montesinos knots with homology sphere double branched cover are slice?
◮ Casson-Harer’s list is incomplete: I found that the Montesinos knot
corresponding to Σ(2, 7, 19) is ribbon.
◮ For 4 or more parameters, if the above conjecture is true then none
are ribbon. Q: Are there further obstructions to a ZHS bounding a Mazur manifold (equiv. being surgery on a knot in S1 × S2)?
◮ A corollary of Taubes theorem on end-periodic manifolds is that
Σ(2, 3, 5)#Σ(2, 3, 5) cannot bound a contractible manifold.
◮ However, this manifold embeds in S4.
Q: Give an example of a ZHS3 which embeds in a ZHS4 but not in S4. Q: Does Σ(3, 5, 8) embed in S4?
15
SLIDE 127
Questions and final remarks
Q: What about embedding Σ(a1, . . . , an), n ≥ 4?
◮ Conjecture: (Kollar) None bound acyclic manifolds, in particular
none embed in S4.
◮ Related to the Montgomery-Yang conjecture.
Q: Which Montesinos knots with homology sphere double branched cover are slice?
◮ Casson-Harer’s list is incomplete: I found that the Montesinos knot
corresponding to Σ(2, 7, 19) is ribbon.
◮ For 4 or more parameters, if the above conjecture is true then none
are ribbon. Q: Are there further obstructions to a ZHS bounding a Mazur manifold (equiv. being surgery on a knot in S1 × S2)?
◮ A corollary of Taubes theorem on end-periodic manifolds is that
Σ(2, 3, 5)#Σ(2, 3, 5) cannot bound a contractible manifold.
◮ However, this manifold embeds in S4.
Q: Give an example of a ZHS3 which embeds in a ZHS4 but not in S4. Q: Does Σ(3, 5, 8) embed in S4?
15
SLIDE 128
Questions and final remarks
Q: What about embedding Σ(a1, . . . , an), n ≥ 4?
◮ Conjecture: (Kollar) None bound acyclic manifolds, in particular
none embed in S4.
◮ Related to the Montgomery-Yang conjecture.
Q: Which Montesinos knots with homology sphere double branched cover are slice?
◮ Casson-Harer’s list is incomplete: I found that the Montesinos knot
corresponding to Σ(2, 7, 19) is ribbon.
◮ For 4 or more parameters, if the above conjecture is true then none
are ribbon. Q: Are there further obstructions to a ZHS bounding a Mazur manifold (equiv. being surgery on a knot in S1 × S2)?
◮ A corollary of Taubes theorem on end-periodic manifolds is that
Σ(2, 3, 5)#Σ(2, 3, 5) cannot bound a contractible manifold.
◮ However, this manifold embeds in S4.
Q: Give an example of a ZHS3 which embeds in a ZHS4 but not in S4. Q: Does Σ(3, 5, 8) embed in S4?
15
SLIDE 129