Which lens spaces embed in S 4 ? Proposition L ( p , q ) , | p | > 1 , does not embed in S 4 . This is a consequence of the following: Proposition (Hantzsche, 1938) If M 3 embeds in S 4 then tor ( H 1 ( M )) ∼ = G ⊕ G for some finite abelian group G. ∼ = Idea: 0 = H 2 ( S 4 ) → H 1 ( M ) → H 1 ( V 1 ) ⊕ H 1 ( V 2 ) → H 1 ( S 4 ) = 0 Notice that H 1 ( L ( p , q )) = Z | p | � = G ⊕ G for | p | > 1. Theorem (Epstein 1965, Zeeman) B 3 , p > 1 embeds in S 4 if and only if p is odd. L ( p , q ) \ ˚ Theorem (Fintushel-Stern, 1985) If L ( p , q )# L ( p , q ′ ) embeds in S 4 , 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

Which lens spaces embed in S 4 ? Proposition L ( p , q ) , | p | > 1 , does not embed in S 4 . This is a consequence of the following: Proposition (Hantzsche, 1938) If M 3 embeds in S 4 then tor ( H 1 ( M )) ∼ = G ⊕ G for some finite abelian group G. ∼ = Idea: 0 = H 2 ( S 4 ) → H 1 ( M ) → H 1 ( V 1 ) ⊕ H 1 ( V 2 ) → H 1 ( S 4 ) = 0 Notice that H 1 ( L ( p , q )) = Z | p | � = G ⊕ G for | p | > 1. Theorem (Epstein 1965, Zeeman) B 3 , p > 1 embeds in S 4 if and only if p is odd. L ( p , q ) \ ˚ Theorem (Fintushel-Stern, 1985) If L ( p , q )# L ( p , q ′ ) embeds in S 4 , 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

Which lens spaces embed in S 4 ? Proposition L ( p , q ) , | p | > 1 , does not embed in S 4 . This is a consequence of the following: Proposition (Hantzsche, 1938) If M 3 embeds in S 4 then tor ( H 1 ( M )) ∼ = G ⊕ G for some finite abelian group G. ∼ = Idea: 0 = H 2 ( S 4 ) → H 1 ( M ) → H 1 ( V 1 ) ⊕ H 1 ( V 2 ) → H 1 ( S 4 ) = 0 Notice that H 1 ( L ( p , q )) = Z | p | � = G ⊕ G for | p | > 1. Theorem (Epstein 1965, Zeeman) B 3 , p > 1 embeds in S 4 if and only if p is odd. L ( p , q ) \ ˚ Theorem (Fintushel-Stern, 1985) If L ( p , q )# L ( p , q ′ ) embeds in S 4 , 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

Embedding homology spheres Theorem (Freedman, 1982) Every Z -homology sphere topologically locally flatly embeds in S 4 . The question of smooth embeddings is much more subtle. For example: Σ(2 , 3 , 5) does not smoothly embed in S 4 : Lemma If a Z -homology sphere M 3 embeds in S 4 then it separates S 4 into two Z -homology B 4 ’s V 1 and V 2 . In particular M bounds an acyclic manifold (a Z -homology B 4 ). Sketch: H i +1 ( S 4 ) → H i ( M ) → H i ( V 1 ) ⊕ H i ( V 2 ) → H i ( S 4 ). Theorem (Rokhlin, 1952) If X 4 is a closed smooth 4 -mfd with H 1 ( X 4 ) = 0 and even intersection form then σ ( X ) / 8 = 0 (mod 2) . ◮ Σ(2 , 3 , 5) bounds the E 8 plumbing W 4 . ◮ If Σ(2 , 3 , 5) embeds in S 4 then V 1 ∪ ∂ W satisfies conditions of Rokhlin’s theorem. ◮ However, σ ( V 1 ∪ W ) / 8 = − 8 / 8 = 1 (mod 2), a contradiction. 5

Embedding homology spheres Theorem (Freedman, 1982) Every Z -homology sphere topologically locally flatly embeds in S 4 . The question of smooth embeddings is much more subtle. For example: Σ(2 , 3 , 5) does not smoothly embed in S 4 : Lemma If a Z -homology sphere M 3 embeds in S 4 then it separates S 4 into two Z -homology B 4 ’s V 1 and V 2 . In particular M bounds an acyclic manifold (a Z -homology B 4 ). Sketch: H i +1 ( S 4 ) → H i ( M ) → H i ( V 1 ) ⊕ H i ( V 2 ) → H i ( S 4 ). Theorem (Rokhlin, 1952) If X 4 is a closed smooth 4 -mfd with H 1 ( X 4 ) = 0 and even intersection form then σ ( X ) / 8 = 0 (mod 2) . ◮ Σ(2 , 3 , 5) bounds the E 8 plumbing W 4 . ◮ If Σ(2 , 3 , 5) embeds in S 4 then V 1 ∪ ∂ W satisfies conditions of Rokhlin’s theorem. ◮ However, σ ( V 1 ∪ W ) / 8 = − 8 / 8 = 1 (mod 2), a contradiction. 5

Embedding homology spheres Theorem (Freedman, 1982) Every Z -homology sphere topologically locally flatly embeds in S 4 . The question of smooth embeddings is much more subtle. For example: Σ(2 , 3 , 5) does not smoothly embed in S 4 : Lemma If a Z -homology sphere M 3 embeds in S 4 then it separates S 4 into two Z -homology B 4 ’s V 1 and V 2 . In particular M bounds an acyclic manifold (a Z -homology B 4 ). Sketch: H i +1 ( S 4 ) → H i ( M ) → H i ( V 1 ) ⊕ H i ( V 2 ) → H i ( S 4 ). Theorem (Rokhlin, 1952) If X 4 is a closed smooth 4 -mfd with H 1 ( X 4 ) = 0 and even intersection form then σ ( X ) / 8 = 0 (mod 2) . ◮ Σ(2 , 3 , 5) bounds the E 8 plumbing W 4 . ◮ If Σ(2 , 3 , 5) embeds in S 4 then V 1 ∪ ∂ W satisfies conditions of Rokhlin’s theorem. ◮ However, σ ( V 1 ∪ W ) / 8 = − 8 / 8 = 1 (mod 2), a contradiction. 5

Obstructing Seifert fibered homology spheres From this point onwards all embeddings are smooth . Let M = Σ( a 1 , . . . , a n ) , a i > 1 pairwise coprime be a Seifert fibered Z HS . 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 spin c 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 S 4 . 6

Obstructing Seifert fibered homology spheres From this point onwards all embeddings are smooth . Let M = Σ( a 1 , . . . , a n ) , a i > 1 pairwise coprime be a Seifert fibered Z HS . 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 spin c 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 S 4 . 6

Obstructing Seifert fibered homology spheres From this point onwards all embeddings are smooth . Let M = Σ( a 1 , . . . , a n ) , a i > 1 pairwise coprime be a Seifert fibered Z HS . 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 spin c 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 S 4 . 6

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 SWFH conn 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 obstructions to bounding an acyclic manifold. 7

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 SWFH conn 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 obstructions to bounding an acyclic manifold. 7

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 SWFH conn 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 obstructions to bounding an acyclic manifold. 7

Mazur manifolds Theorem (Akbulut-Kirby, 1978) Σ(3 , 4 , 5) embeds in S 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 S 4 , hence W embeds in S 4 . 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) = S 3 × S 1 . ◮ Can unknot γ , so γ isotopic to pt × S 1 ⊂ S 3 × S 1 . ◮ Hence, can cancel 1-h and 2-h, so DW = ∂ (0-h) = S 4 . 8

Mazur manifolds Theorem (Akbulut-Kirby, 1978) Σ(3 , 4 , 5) embeds in S 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 S 4 , hence W embeds in S 4 . 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) = S 3 × S 1 . ◮ Can unknot γ , so γ isotopic to pt × S 1 ⊂ S 3 × S 1 . ◮ Hence, can cancel 1-h and 2-h, so DW = ∂ (0-h) = S 4 . 8

Mazur manifolds 4 Theorem (Akbulut-Kirby, 1978) Σ(3 , 4 , 5) embeds in S 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 S 4 , hence W embeds in S 4 . 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) = S 3 × S 1 . ◮ Can unknot γ , so γ isotopic to pt × S 1 ⊂ S 3 × S 1 . ◮ Hence, can cancel 1-h and 2-h, so DW = ∂ (0-h) = S 4 . 8

Mazur manifolds 4 Theorem (Akbulut-Kirby, 1978) Σ(3 , 4 , 5) embeds in S 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 S 4 , hence W embeds in S 4 . 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) = S 3 × S 1 . ◮ Can unknot γ , so γ isotopic to pt × S 1 ⊂ S 3 × S 1 . ◮ Hence, can cancel 1-h and 2-h, so DW = ∂ (0-h) = S 4 . 8

Mazur manifolds 4 Theorem (Akbulut-Kirby, 1978) Σ(3 , 4 , 5) embeds in S 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 S 4 , hence W embeds in S 4 . 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) = S 3 × S 1 . ◮ Can unknot γ , so γ isotopic to pt × S 1 ⊂ S 3 × S 1 . ◮ Hence, can cancel 1-h and 2-h, so DW = ∂ (0-h) = S 4 . 8

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 M 3 be a Z HS which is the double branched cover of a knot K in S 3 . If K is ribbon via a single band move then M 3 is the boundary of a Mazur manifold. Example: Σ(3 , 4 , 5) is the double branched cover of the Montesinos knot: 9

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 M 3 be a Z HS which is the double branched cover of a knot K in S 3 . If K is ribbon via a single band move then M 3 is the boundary of a Mazur manifold. Example: Σ(3 , 4 , 5) is the double branched cover of the Montesinos knot: 9

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 M 3 be a Z HS which is the double branched cover of a knot K in S 3 . If K is ribbon via a single band move then M 3 is the boundary of a Mazur manifold. Example: Σ(3 , 4 , 5) is the double branched cover of the Montesinos knot: 9

I’ll sketch a proof of Casson-Harer’s theorem. Lemma 1 Proof: 10

I’ll sketch a proof of Casson-Harer’s theorem. Lemma 1 branched double cover Proof: 10

I’ll sketch a proof of Casson-Harer’s theorem. Lemma 1 branched double cover Proof: π rotation 10

I’ll sketch a proof of Casson-Harer’s theorem. Lemma 1 branched double cover Proof: π rotation fold fold 10

I’ll sketch a proof of Casson-Harer’s theorem. Lemma 1 branched double cover Proof: π rotation = fold fold 10

Lemma 2 S 1 × S 2 is the double branched cover of the 2 component unlink in S 3 . Proof: ◮ S 3 = B 1 ∪ B 2 ◮ Passing to double branched covers: Σ 2 ( S 3 ) = Σ 2 ( B 1 ) ∪ Σ 2 ( B 2 ). ◮ Σ 2 ( S 3 ) is two solid tori glued along boundaries by identity map. ◮ Σ 2 ( S 3 ) = S 1 × S 2 . 11

Lemma 2 S 1 × S 2 is the double branched cover of the 2 component unlink in S 3 . Proof: ◮ S 3 = B 1 ∪ B 2 ◮ Passing to double branched covers: Σ 2 ( S 3 ) = Σ 2 ( B 1 ) ∪ Σ 2 ( B 2 ). ◮ Σ 2 ( S 3 ) is two solid tori glued along boundaries by identity map. ◮ Σ 2 ( S 3 ) = S 1 × S 2 . 11

Lemma 2 S 1 × S 2 is the double branched cover of the 2 component unlink in S 3 . Proof: ◮ S 3 = B 1 ∪ B 2 ◮ Passing to double branched covers: Σ 2 ( S 3 ) = Σ 2 ( B 1 ) ∪ Σ 2 ( B 2 ). ◮ Σ 2 ( S 3 ) is two solid tori glued along boundaries by identity map. ◮ Σ 2 ( S 3 ) = S 1 × S 2 . B 2 B 1 11

Lemma 2 S 1 × S 2 is the double branched cover of the 2 component unlink in S 3 . Proof: ◮ S 3 = B 1 ∪ B 2 ◮ Passing to double branched covers: Σ 2 ( S 3 ) = Σ 2 ( B 1 ) ∪ Σ 2 ( B 2 ). ◮ Σ 2 ( S 3 ) is two solid tori glued along boundaries by identity map. ◮ Σ 2 ( S 3 ) = S 1 × S 2 . B 2 B 1 11

Lemma 2 S 1 × S 2 is the double branched cover of the 2 component unlink in S 3 . Proof: ◮ S 3 = B 1 ∪ B 2 ◮ Passing to double branched covers: Σ 2 ( S 3 ) = Σ 2 ( B 1 ) ∪ Σ 2 ( B 2 ). ◮ Σ 2 ( S 3 ) is two solid tori glued along boundaries by identity map. ◮ Σ 2 ( S 3 ) = S 1 × S 2 . glue by identity B 2 branched double cover B 1 S 1 x S 2 11

Lemma 2 S 1 × S 2 is the double branched cover of the 2 component unlink in S 3 . Proof: ◮ S 3 = B 1 ∪ B 2 ◮ Passing to double branched covers: Σ 2 ( S 3 ) = Σ 2 ( B 1 ) ∪ Σ 2 ( B 2 ). ◮ Σ 2 ( S 3 ) is two solid tori glued along boundaries by identity map. ◮ Σ 2 ( S 3 ) = S 1 × S 2 . glue by identity B 2 branched double cover B 1 S 1 x S 2 11

Theorem (Casson-Harer) Let M 3 be a Z HS which is the double branched cover of a knot K in S 3 . If K is ribbon via a single band move then M 3 is the boundary of a Mazur manifold. Proof: ◮ Claim: M 3 is surgery on a knot in S 1 × S 2 . ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K . ◮ Downstairs: Replace ( B 3 , 2-arcs) with another ( B 3 , 2-arcs). ◮ Upstairs: Replace solid torus in S 1 × S 2 with another solid torus. Change 0-framed unknot to dotted circle to get Mazur manifold. 12

Theorem (Casson-Harer) Let M 3 be a Z HS which is the double branched cover of a knot K in S 3 . If K is ribbon via a single band move then M 3 is the boundary of a Mazur manifold. Proof: ◮ Claim: M 3 is surgery on a knot in S 1 × S 2 . ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K . ◮ Downstairs: Replace ( B 3 , 2-arcs) with another ( B 3 , 2-arcs). ◮ Upstairs: Replace solid torus in S 1 × S 2 with another solid torus. Change 0-framed unknot to dotted circle to get Mazur manifold. 12

Theorem (Casson-Harer) Let M 3 be a Z HS which is the double branched cover of a knot K in S 3 . If K is ribbon via a single band move then M 3 is the boundary of a Mazur manifold. Proof: ◮ Claim: M 3 is surgery on a knot in S 1 × S 2 . ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K . ◮ Downstairs: Replace ( B 3 , 2-arcs) with another ( B 3 , 2-arcs). ◮ Upstairs: Replace solid torus in S 1 × S 2 with another solid torus. Change 0-framed unknot to dotted circle to get Mazur manifold. 12

Theorem (Casson-Harer) Let M 3 be a Z HS which is the double branched cover of a knot K in S 3 . If K is ribbon via a single band move then M 3 is the boundary of a Mazur manifold. Proof: ◮ Claim: M 3 is surgery on a knot in S 1 × S 2 . ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K . ◮ Downstairs: Replace ( B 3 , 2-arcs) with another ( B 3 , 2-arcs). ◮ Upstairs: Replace solid torus in S 1 × S 2 with another solid torus. Change 0-framed unknot to dotted circle to get Mazur manifold. canceling band move 12

Theorem (Casson-Harer) Let M 3 be a Z HS which is the double branched cover of a knot K in S 3 . If K is ribbon via a single band move then M 3 is the boundary of a Mazur manifold. Proof: ◮ Claim: M 3 is surgery on a knot in S 1 × S 2 . ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K . ◮ Downstairs: Replace ( B 3 , 2-arcs) with another ( B 3 , 2-arcs). ◮ Upstairs: Replace solid torus in S 1 × S 2 with another solid torus. Change 0-framed unknot to dotted circle to get Mazur manifold. B 3 12

Theorem (Casson-Harer) Let M 3 be a Z HS which is the double branched cover of a knot K in S 3 . If K is ribbon via a single band move then M 3 is the boundary of a Mazur manifold. Proof: ◮ Claim: M 3 is surgery on a knot in S 1 × S 2 . ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K . ◮ Downstairs: Replace ( B 3 , 2-arcs) with another ( B 3 , 2-arcs). ◮ Upstairs: Replace solid torus in S 1 × S 2 with another solid torus. Change 0-framed unknot to dotted circle to get Mazur manifold. B 3 remove this ball (torus in double branched cover) 12

Theorem (Casson-Harer) Let M 3 be a Z HS which is the double branched cover of a knot K in S 3 . If K is ribbon via a single band move then M 3 is the boundary of a Mazur manifold. Proof: ◮ Claim: M 3 is surgery on a knot in S 1 × S 2 . ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K . ◮ Downstairs: Replace ( B 3 , 2-arcs) with another ( B 3 , 2-arcs). ◮ Upstairs: Replace solid torus in S 1 × S 2 with another solid torus. Change 0-framed unknot to dotted circle to get Mazur manifold. B 3 glue in this ball (torus in double branched cover) 12

Theorem (Casson-Harer) Let M 3 be a Z HS which is the double branched cover of a knot K in S 3 . If K is ribbon via a single band move then M 3 is the boundary of a Mazur manifold. Proof: ◮ Claim: M 3 is surgery on a knot in S 1 × S 2 . ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K . ◮ Downstairs: Replace ( B 3 , 2-arcs) with another ( B 3 , 2-arcs). ◮ Upstairs: Replace solid torus in S 1 × S 2 with another solid torus. Change 0-framed unknot to dotted circle to get Mazur manifold. 4 B 3 0 glue in this ball (torus in double branched cover) 12

Theorem (Casson-Harer) Let M 3 be a Z HS which is the double branched cover of a knot K in S 3 . If K is ribbon via a single band move then M 3 is the boundary of a Mazur manifold. Proof: ◮ Claim: M 3 is surgery on a knot in S 1 × S 2 . ◮ A single band move on K gives the unlink. ◮ Hence, a single band move on the unlink gives K . ◮ Downstairs: Replace ( B 3 , 2-arcs) with another ( B 3 , 2-arcs). ◮ Upstairs: Replace solid torus in S 1 × S 2 with another solid torus. Change 0-framed unknot to dotted circle to get Mazur manifold. 4 B 3 glue in this ball (torus in double branched cover) 12

All Σ( p , q , r ) known to bound a Z HB 4 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 Z HB 4 : ◮ (Fintushel-Stern) Σ(2 , 7 , 19), Σ(3 , 5 , 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2 , 7 , 47), Σ(3 , 5 , 49) bound Z HB 4 ’s. ◮ (Fickle) Σ(2 , 3 , 25) bounds a Mazur manifold. Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2 , 3 and k even bound Z HB 4 ’s. 13

All Σ( p , q , r ) known to bound a Z HB 4 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 Z HB 4 : ◮ (Fintushel-Stern) Σ(2 , 7 , 19), Σ(3 , 5 , 19) bound Mazur manifolds. ◮ (Fintushel-Stern) Σ(2 , 7 , 47), Σ(3 , 5 , 49) bound Z HB 4 ’s. ◮ (Fickle) Σ(2 , 3 , 25) bounds a Mazur manifold. Conjecture (Fintushel-Stern): Brieskorn spheres with p = 2 , 3 and k even bound Z HB 4 ’s. 13

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