Prismatic Maps for the Topological Tverberg Conjecture Isaac - - PowerPoint PPT Presentation

Prismatic Maps for the Topological Tverberg Conjecture

Isaac Mabillard Joint work with Uli Wagner

Geometry → Algebra

Geometry → Algebra Hope ∼ =

General Problem:

Let K be a simplicial complex and r ≥ 2. Does there exist a continuous map f : K → Rd without r-fold intersections?

General Problem:

Let K be a simplicial complex and r ≥ 2. Does there exist a continuous map f : K → Rd without r-fold intersections? A point p ∈ Rd is an r-fold intersection if there exit x1, ..., xr ∈ |K| distinct such that p = fx1 = · · · = fxr K f(K)

f

− → x1 x2 p

General Problem:

Let K be a simplicial complex and r ≥ 2. Does there exist a continuous map f : K → Rd without r-fold intersections? A point p ∈ Rd is an r-fold intersection if there exit x1, ..., xr ∈ |K| distinct such that p = fx1 = · · · = fxr A map f : K → Rd without r-fold intersection is called r-embedding K f(K)

f

− → x1 x2 p

Example: f : K2 → R3

Example: f : K2 → R3

K = real projective plane RP 2

Example: f : K2 → R3

K = real projective plane RP 2 RP 2 Boy’s Surface

f

− →

Example: f : K2 → R3

K = real projective plane RP 2 RP 2 Boy’s Surface

f

− → 2-fold intersection

Example: f : K2 → R3

K = real projective plane RP 2 RP 2 Boy’s Surface

f

− → 2-fold intersection (unique) 3-fold intersection

Example: f : K2 → R3

K = real projective plane RP 2 RP 2 Boy’s Surface

f

− → 2-fold intersection (unique) 3-fold intersection f : RP 2 → R3 is a 4-embedding (no 4-fold intersections)

Classical Case: Maps without 2-fold intersections

Classical Case: Maps without 2-fold intersections

Goal: Find f : K → Rd continuous & injective (i.e.,f is an embedding)

Classical Case: Maps without 2-fold intersections

Goal: Find f : K → Rd continuous & injective (i.e.,f is an embedding) Theorem (van Kampen–Shapiro–Wu): ∃f : Km ֒ → R2m ⇔ ∃ f : K×2

δ

→S2 S2m−1 provided m = 2.

Classical Case: Maps without 2-fold intersections

Goal: Find f : K → Rd continuous & injective (i.e.,f is an embedding) Theorem (van Kampen–Shapiro–Wu): ∃f : Km ֒ → R2m ⇔ ∃ f : K×2

δ

→S2 S2m−1 provided m = 2. ‘easy’ Proposition The existence of K×2

δ

→S2 S2m−1 is algorithmically solvable.

Classical Case: Maps without 2-fold intersections

Goal: Find f : K → Rd continuous & injective (i.e.,f is an embedding) Theorem (van Kampen–Shapiro–Wu): ∃f : Km ֒ → R2m ⇔ ∃ f : K×2

δ

→S2 S2m−1 provided m = 2. ‘easy’ Proposition The existence of K×2

δ

→S2 S2m−1 is algorithmically solvable.

• Corollary. The existence of an embedding Km ֒

→ R2m is algorithmically solvable, provided m = 2.

What about maps without r-fold intersections?

What about maps without r-fold intersections?

Goal: Find f : K → Rd continuous & without r-fold intersection (i.e.,f is an r-embedding) r = 3 f(K) ⊂ R3

What about maps without r-fold intersections?

Goal: Find f : K → Rd continuous & without r-fold intersection (i.e.,f is an r-embedding) An necessary condition for the existence of f: 1) Define the r-fold deleted product of K: K×r

δ

:= {σ1 × · · · × σr | σi ∈ K and σi ∩ σj = ∅} ⊂ K×r r = 3 f(K) ⊂ R3

What about maps without r-fold intersections?

Goal: Find f : K → Rd continuous & without r-fold intersection (i.e.,f is an r-embedding) An necessary condition for the existence of f: 1) Define the r-fold deleted product of K: K×r

δ

:= {σ1 × · · · × σr | σi ∈ K and σi ∩ σj = ∅} ⊂ K×r 2) Given an r-embedding f : K → Rd, define

• f :

K×r

δ

→ Rd×r (x1, . . . , xr) → (fx1, . . . , fxr) r = 3 f(K) ⊂ R3

• f :

K×r

δ

→ Rd×r (x1, . . . , xr) → (fx1, . . . , fxr)

Two properties of f

• f :

K×r

δ

→ Rd×r (x1, . . . , xr) → (fx1, . . . , fxr) A) The symmetric group Sr acts on both K×r

δ

and Rd×r by permutation of the coordinates

• f is compatible with both actions (i.e.,

f is Sr-equivariant): For all ρ ∈ Sr

• f ◦ ρ = ρ ◦

f

Two properties of f

• f :

K×r

δ

→ Rd×r (x1, . . . , xr) → (fx1, . . . , fxr) B) (xi ∈ σi ∈ K and σi ∩ σj = ∅) ⇒ all the xi are distinct f is an r-embedding ⇒ ¬(fx1 = · · · = fxr) A) The symmetric group Sr acts on both K×r

δ

and Rd×r by permutation of the coordinates

• f is compatible with both actions (i.e.,

f is Sr-equivariant): For all ρ ∈ Sr

• f ◦ ρ = ρ ◦

f

Two properties of f

• f :

K×r

δ

→ Rd×r (x1, . . . , xr) → (fx1, . . . , fxr) B) (xi ∈ σi ∈ K and σi ∩ σj = ∅) ⇒ all the xi are distinct f is an r-embedding ⇒ ¬(fx1 = · · · = fxr) Hence:

• f : K×r

δ

→Sr Rd×r\{(x, . . . , x) | x ∈ Rd} A) The symmetric group Sr acts on both K×r

δ

and Rd×r by permutation of the coordinates

• f is compatible with both actions (i.e.,

f is Sr-equivariant): For all ρ ∈ Sr

• f ◦ ρ = ρ ◦

f

Two properties of f

• f :

K×r

δ

→ Rd×r (x1, . . . , xr) → (fx1, . . . , fxr) B) (xi ∈ σi ∈ K and σi ∩ σj = ∅) ⇒ all the xi are distinct f is an r-embedding ⇒ ¬(fx1 = · · · = fxr) Hence:

• f : K×r

δ

→Sr Rd×r\{(x, . . . , x) | x ∈ Rd} A) The symmetric group Sr acts on both K×r

δ

and Rd×r by permutation of the coordinates

• f is compatible with both actions (i.e.,

f is Sr-equivariant): For all ρ ∈ Sr

• f ◦ ρ = ρ ◦

f

Two properties of f

≃ S(r−1)d−1

f : Km → Rd such that for all σ1, . . . , σr ∈ K with σi ∩ σj = ∅ fσ1 ∩ · · · ∩ fσr = ∅

f : Km → Rd such that for all σ1, . . . , σr ∈ K with σi ∩ σj = ∅ fσ1 ∩ · · · ∩ fσr = ∅ ⇓ ∃ f : K×r

δ

→Sr S(r−1)d−1

f : Km → Rd such that for all σ1, . . . , σr ∈ K with σi ∩ σj = ∅ fσ1 ∩ · · · ∩ fσr = ∅ ⇓ ∃ f : K×r

δ

→Sr S(r−1)d−1 ⇑

?

f : Km → Rd such that for all σ1, . . . , σr ∈ K with σi ∩ σj = ∅ fσ1 ∩ · · · ∩ fσr = ∅ ⇓ ∃ f : K×r

δ

→Sr S(r−1)d−1 ⇑ provided m = (r − 1)k, d = rk and k ≥ 3 yes!

f : Km → Rd such that for all σ1, . . . , σr ∈ K with σi ∩ σj = ∅ fσ1 ∩ · · · ∩ fσr = ∅ ⇓ ∃ f : K×r

δ

→Sr S(r−1)d−1 ⇑ provided m = (r − 1)k, d = rk and k ≥ 3 f is an almost r-embedding yes!

Theorem: provided k ≥ 3. ∃f : K(r−1)k → Rrk almost r-embedding ⇔ ∃ f : K×r

δ

→Sr S(r−1)rk−1

Theorem: provided k ≥ 3. geometric problem algebraic problem ⇔ (map without intersection) (equivariant map) ∃f : K(r−1)k → Rrk almost r-embedding ⇔ ∃ f : K×r

δ

→Sr S(r−1)rk−1

Theorem: provided k ≥ 3. geometric problem algebraic problem ⇔ (map without intersection) (equivariant map) easy Proposition The existence of K×r

δ

→Sr S(r−1)rk−1 is algorithmically solvable. ∃f : K(r−1)k → Rrk almost r-embedding ⇔ ∃ f : K×r

δ

→Sr S(r−1)rk−1

Theorem: provided k ≥ 3. geometric problem algebraic problem ⇔ (map without intersection) (equivariant map) easy Proposition The existence of K×r

δ

→Sr S(r−1)rk−1 is algorithmically solvable.

• Corollary. The existence of f : K(r−1)k → Rrk almost

r-embedding is algorithmically solvable, provided k ≥ 3. ∃f : K(r−1)k → Rrk almost r-embedding ⇔ ∃ f : K×r

δ

→Sr S(r−1)rk−1

Our Main Tool: an r-fold analogue of the Whitney Trick

Our Main Tool: an r-fold analogue of the Whitney Trick Classical Whitney Trick with two balls σp and τ q in Rp+q:

σp τ q p, q ≥ 3 Rp+q x y

Our Main Tool: an r-fold analogue of the Whitney Trick Classical Whitney Trick with two balls σp and τ q in Rp+q:

σp τ q p, q ≥ 3 Rp+q x +1 y −1

Our Main Tool: an r-fold analogue of the Whitney Trick Classical Whitney Trick with two balls σp and τ q in Rp+q:

σp τ q p, q ≥ 3 Rp+q

Whitney Disk D2

x +1 y −1

Our Main Tool: an r-fold analogue of the Whitney Trick Classical Whitney Trick with two balls σp and τ q in Rp+q:

σp τ q p, q ≥ 3 Rp+q

Whitney Disk D2

x +1 y −1 push σp along the Whitney Disk

• σp

What happens with more than two balls?

x y +1 −1 σ6 τ 6 µ6 R9

What happens with more than two balls?

x y +1 −1 σ6 R9 σ ∩ µ σ ∩ τ

What happens with more than two balls?

x y +1 −1 σ6 R9 σ ∩ µ σ ∩ τ

• σ ∩ µ

Whitney trick for two balls

What happens with more than two balls?

x y +1 −1 σ6 R9 σ ∩ µ σ ∩ τ

• σ ∩ µ

Whitney trick for two balls Then, use that σ6 is “flat” (codimension ≥ 3) to extend the solution to R9.

What happens with more than two balls?

x y +1 −1 σ6 R9 σ ∩ µ σ ∩ τ

• σ ∩ µ

Whitney trick for two balls Then, use that σ6 is “flat” (codimension ≥ 3) to extend the solution to R9. Problem: σ ∩ τ and σ ∩ µ are, in general, not connected spaces x y +1 −1 σ σ ∩ τ σ ∩ µ

Piping + Unpiping Trick

τ p Rp+3

Piping + Unpiping Trick

τ p Rp+3

Piping + Unpiping Trick

τ p Rp+3

Piping + Unpiping Trick

τ p Rp+3 1-handle

Piping + Unpiping Trick

τ p Rp+3 1-handle

Piping + Unpiping Trick

τ p Rp+3 1-handle

Piping + Unpiping Trick

τ p Rp+3 1-handle 2-handle

Piping + Unpiping Trick

τ p Rp+3 1-handle 2-handle ∼ = τ p + the two handles ∼ = a p-ball

Back to the intersection problem

σ τ

Back to the intersection problem

1-handle σ τ

Back to the intersection problem

1-handle complementary 2-handle σ τ

Back to the intersection problem

1-handle on σ ∩ τ Hence, we can add 1-handles on σ ∩ τ. I.e., we can make σ ∩ τ connected. 1-handle σ

3-fold Whitney Trick

τ µ σ x y +1 −1 y x σ ∩ τ σ ∩ µ σ We can assume σ ∩ τ and σ ∩ µ are connected. Hence we can use the classical Whitney trick to solve the 3-balls situation, i.e., to remove triple intersection points.

r-fold Whitney Trick

Given r balls B1, · · · , Br mapped by a f into Rd in general position f : B1 ⊔ · · · ⊔ Br → Rd with d − dim(Bi) ≥ 3 and

• i

d − dim(Bi) = d. If f(B1) ∩ · · · ∩ f(Br) = {x, y} two points of opposite signs. Then we can remove these two points by a move along a 2-dimensional cone (≈“Whitney disk”). In particular, we can avoid any codimension ≥ 3 object in Rd during this move.

classical Whitney Trick ⇒ first part of Van Kampen Embeddability (k = 2): Kk → R2k almost 2-embeds ⇔ K×2

δ

→S2 S2k−1

classical Whitney Trick ⇒ first part of Van Kampen Embeddability (k = 2): Kk → R2k almost 2-embeds ⇔ K×2

δ

→S2 S2k−1 r-fold Whitney Trick ⇒ For r, k ≥ 3, K(r−1)k → Rrk almost r-embeds ⇔ K×r

δ

→Sr Sr(r−1)k−1

classical Whitney Trick ⇒ first part of Van Kampen Embeddability (k = 2): Kk → R2k almost 2-embeds ⇔ K×2

δ

→S2 S2k−1 r-fold Whitney Trick ⇒ For r, k ≥ 3, K(r−1)k → Rrk almost r-embeds ⇔ K×r

δ

→Sr Sr(r−1)k−1 ⇔ check a system of linear equations over Z

Application: Topological Tverberg

Topological Tverberg Conjecture: Given r, d ≥ 2, there exists no almost r-embedding ∆(r−1)(d+1) → Rd. → f ∆3 σ1 σ2 Example for r = 2 fσ1 ∩ fσ2 = ∅ Hence f is not an almost 2-embedding R2

Application: Topological Tverberg

Topological Tverberg Conjecture: Given r, d ≥ 2, there exists no almost r-embedding ∆(r−1)(d+1) → Rd. → f ∆3 σ1 σ2 Example for r = 2 fσ1 ∩ fσ2 = ∅ Hence f is not an almost 2-embedding R2 The conjecture holds for r = primepower (Ozaydin87)

Application: Topological Tverberg

Topological Tverberg Conjecture: Given r, d ≥ 2, there exists no almost r-embedding ∆(r−1)(d+1) → Rd. (Ozaydin 1987) If r is not a prime power and dim X ≤ d(r − 1) with free Sr-action, then X →Sr S(r−1)d−1

Application: Topological Tverberg

Topological Tverberg Conjecture: Given r, d ≥ 2, there exists no almost r-embedding ∆(r−1)(d+1) → Rd. (Ozaydin 1987) If r is not a prime power and dim X ≤ d(r − 1) with free Sr-action, then X →Sr S(r−1)d−1 (M-Wagner) Provided k ≥ 3: K×r

δ

→Sr S(r−1)rk−1 ⇔ K(r−1)k → Rrk almost r-embeds

Application: Topological Tverberg

Topological Tverberg Conjecture: Given r, d ≥ 2, there exists no almost r-embedding ∆(r−1)(d+1) → Rd. (Ozaydin 1987) If r is not a prime power and dim X ≤ d(r − 1) with free Sr-action, then X →Sr S(r−1)d−1 (Gromov 2010, Blagojevic-Frick-Ziegler 2014) ∆(3r+2)(r−1)

≤3(r−1)

→ R3r almost r-embeds ⇒∆(3r+2)(r−1) → R3r+1 almost r-embeds. (M-Wagner) Provided k ≥ 3: K×r

δ

→Sr S(r−1)rk−1 ⇔ K(r−1)k → Rrk almost r-embeds

Application: Topological Tverberg

Topological Tverberg Conjecture: Given r, d ≥ 2, there exists no almost r-embedding ∆(r−1)(d+1) → Rd. (Ozaydin 1987) If r is not a prime power and dim X ≤ d(r − 1) with free Sr-action, then X →Sr S(r−1)d−1 (Gromov 2010, Blagojevic-Frick-Ziegler 2014) ∆(3r+2)(r−1)

≤3(r−1)

→ R3r almost r-embeds ⇒∆(3r+2)(r−1) → R3r+1 almost r-embeds. Frick’s observation: (O) + (MW) + (G) ⇒ there exists an almost 6-embedding ∆100 → R19 (M-Wagner) Provided k ≥ 3: K×r

δ

→Sr S(r−1)rk−1 ⇔ K(r−1)k → Rrk almost r-embeds

Application: Topological Tverberg

Topological Tverberg Conjecture: Given r, d ≥ 2, there exists no almost r-embedding ∆(r−1)(d+1) → Rd. (Gromov 2010, Blagojevic-Frick-Ziegler 2014) ∆(3r+2)(r−1)

≤3(r−1)

→ R3r almost r-embeds ⇒∆(3r+2)(r−1) → R3r+1 almost r-embeds. Frick’s observation: (O) + (MW) + (G) ⇒ there exists an almost 6-embedding ∆100 → R19 (Ozaydin 1987) (∆100

≤15) ×6 δ

→S6 S89 (M-Wagner) Provided k ≥ 3: K×r

δ

→Sr S(r−1)rk−1 ⇔ K(r−1)k → Rrk almost r-embeds

Application: Topological Tverberg

Topological Tverberg Conjecture: Given r, d ≥ 2, there exists no almost r-embedding ∆(r−1)(d+1) → Rd. (M-Wagner) (∆100

≤15) ×6 δ

→S6 S89 ⇒ ∆100

≤15 → R18 almost 6-embeds

(Gromov 2010, Blagojevic-Frick-Ziegler 2014) ∆(3r+2)(r−1)

≤3(r−1)

→ R3r almost r-embeds ⇒∆(3r+2)(r−1) → R3r+1 almost r-embeds. Frick’s observation: (O) + (MW) + (G) ⇒ there exists an almost 6-embedding ∆100 → R19 (Ozaydin 1987) (∆100

≤15) ×6 δ

→S6 S89

Application: Topological Tverberg

Topological Tverberg Conjecture: Given r, d ≥ 2, there exists no almost r-embedding ∆(r−1)(d+1) → Rd. (M-Wagner) (∆100

≤15) ×6 δ

→S6 S89 ⇒ ∆100

≤15 → R18 almost 6-embeds

Frick’s observation: (O) + (MW) + (G) ⇒ there exists an almost 6-embedding ∆100 → R19 (Ozaydin 1987) (∆100

≤15) ×6 δ

→S6 S89 (Gromov 2010, Blagojevic-Frick-Ziegler 2014) ∆100

≤15 → R18 almost 6-embeds

⇒∆100 → R19 almost 6-embeds.

Theorem (Avvakumov-M-Skopenkov-Wagner). For d ≥ 2r and r not a prime power, there exists an almost r-embedding ∆(d+1)(r−1) → Rd

Theorem (Avvakumov-M-Skopenkov-Wagner). For d ≥ 2r and r not a prime power, there exists an almost r-embedding ∆(d+1)(r−1) → Rd Minimal counterexample: almost 6-embedding ∆65 → R12.

Theorem (Avvakumov-M-Skopenkov-Wagner). For d ≥ 2r and r not a prime power, there exists an almost r-embedding ∆(d+1)(r−1) → Rd Minimal counterexample: almost 6-embedding ∆65 → R12. What happens for d ≤ 11?

Theorem (Avvakumov-M-Skopenkov-Wagner). For d ≥ 2r and r not a prime power, there exists an almost r-embedding ∆(d+1)(r−1) → Rd Minimal counterexample: almost 6-embedding ∆65 → R12. What happens for d ≤ 11? First open case of the conjecture: almost 6-embedding ∆15 → R2. I.e., a drawing of K16 without

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

• r

How the minimal counterexample ∆65 → R12 was obtained? Two new tools:

How the minimal counterexample ∆65 → R12 was obtained? Two new tools:

1) Prismatic maps (forcing codimension) ∆8 → ⊂ R3 All the Tverberg partitions are made of triangles

How the minimal counterexample ∆65 → R12 was obtained? Two new tools:

1) Prismatic maps (forcing codimension) ∆8 → ⊂ R3 All the Tverberg partitions are made of triangles 2) A codimension 2 (!) Whitney Trick (Avvakumov-M-Skopenkov-Wagner) Provided k ≥ 2 and r ≥ 3: ∃K(r−1)k → Rrk almost r-embedding ⇔ K×r

δ

→Sr S(r−1)rk−1

Frick’s counterexample ∆100 → R19

19 = 6 · 3 + 1

Frick’s counterexample ∆100 → R19

19 = 6 · 3 + 1

Ozaydin (smallest non-prime power) Gromov trick, Constraint method (BFZ) M-Wagner r-fold Whitney Trick

Ozaydin (smallest non-prime power) prismatic maps

18 = 6 · 3 + 0

M-Wagner prismatic counterexample ∆95 → R18

M-Wagner r-fold Whitney Trick

codim 2 r-fold Whitney Trick Ozaydin (smallest non-prime power) prismatic maps

12 = 6 · 2 + 0

Avvakumov-M-Skopenkov-Wagner prismatic codim 2 counterexample ∆65 → R12

codim 2 r-fold Whitney Trick Ozaydin (smallest non-prime power) prismatic maps

12 = 6 · 2 + 0

Avvakumov-M-Skopenkov-Wagner prismatic codim 2 counterexample ∆65 → R12

What happens in lower dimension (2 ≤ d ≤ 11) remains a mystery...

In another direction...

In another direction...

The van-Kampen-Shaprio-Wu theorem was vastly extended in the 60s

In another direction...

The van-Kampen-Shaprio-Wu theorem was vastly extended in the 60s Theorem (Haefliger-Weber 60’s): ∃f : Km ֒ → Rd ⇔ ∃ f : K×2

δ

→S2 Sd−1 provided 2d ≥ 3m + 3 (=metastable range)

In another direction...

The van-Kampen-Shaprio-Wu theorem was vastly extended in the 60s Theorem (Haefliger-Weber 60’s): ∃f : Km ֒ → Rd ⇔ ∃ f : K×2

δ

→S2 Sd−1 provided 2d ≥ 3m + 3 (=metastable range) Theorem (Cadek-Krcal-Vokrinek 13) The existence of K×2

δ

→S2 Sd−1 is algorithmically solvable.

In another direction...

The van-Kampen-Shaprio-Wu theorem was vastly extended in the 60s Theorem (Haefliger-Weber 60’s): ∃f : Km ֒ → Rd ⇔ ∃ f : K×2

δ

→S2 Sd−1 provided 2d ≥ 3m + 3 (=metastable range) Theorem (Cadek-Krcal-Vokrinek 13) The existence of K×2

δ

→S2 Sd−1 is algorithmically solvable.

• Corollary. The existence of an embedding Km ֒

→ Rd is algorithmically solvable, provided d 1.5m

∃f : Km → Rd almost r-embedding ⇔ ∃ f : K×r

δ

→Sr S(r−1)d−1 Theorem (M-Wagner) provided rd ≥ (r + 1)m + 3 (= r-metastable range).

∃f : Km → Rd almost r-embedding ⇔ ∃ f : K×r

δ

→Sr S(r−1)d−1 Theorem (M-Wagner) provided rd ≥ (r + 1)m + 3 (= r-metastable range). Theorem (Filakovksy-Vokrinek). The existence of K×r

δ

→Sr S(r−1)d−1 is algorithmically solvable.

∃f : Km → Rd almost r-embedding ⇔ ∃ f : K×r

δ

→Sr S(r−1)d−1 Theorem (M-Wagner) provided rd ≥ (r + 1)m + 3 (= r-metastable range). Theorem (Filakovksy-Vokrinek). The existence of K×r

δ

→Sr S(r−1)d−1 is algorithmically solvable.

• Corollary. The existence of f : Km → Rd almost r-embedding

is algorithmically solvable, provided d r+1

r m.

∃f : Km → Rd almost r-embedding ⇔ ∃ f : K×r

δ

→Sr S(r−1)d−1 Theorem (M-Wagner) provided rd ≥ (r + 1)m + 3 (= r-metastable range). Theorem (Filakovksy-Vokrinek). The existence of K×r

δ

→Sr S(r−1)d−1 is algorithmically solvable.

• Corollary. The existence of f : Km → Rd almost r-embedding

is algorithmically solvable, provided d r+1

r m.

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