Homology cobordism and triangulations
Ciprian Manolescu
UCLA
August 3, 2018
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 1 / 20 Question (Poincar´e1899): Does every smooth manifold admit a triangulation?
Answer (Cairns, Whitehead 1940): Yes. Every smooth manifold has an essentially unique piecewise linear (PL) structure,∼ and therefore it is triangulable.
Question (Kneser 1924): Does every topological manifold admit a triangulation? We can ask this about arbitrary triangulations, or about the more natural PL (combinatorial) triangulations.
Triangulations of manifolds
A triangulation of a topological space X is a homeomorphism from X to a simplicial complex.
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 2 / 20 Answer (Cairns, Whitehead 1940): Yes. Every smooth manifold has an essentially unique piecewise linear (PL) structure,∼ and therefore it is triangulable.
Question (Kneser 1924): Does every topological manifold admit a triangulation? We can ask this about arbitrary triangulations, or about the more natural PL (combinatorial) triangulations.
Triangulations of manifolds
A triangulation of a topological space X is a homeomorphism from X to a simplicial complex.
Question (Poincar´e1899): Does every smooth manifold admit a triangulation?
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 2 / 20 Question (Kneser 1924): Does every topological manifold admit a triangulation? We can ask this about arbitrary triangulations, or about the more natural PL (combinatorial) triangulations.
Triangulations of manifolds
A triangulation of a topological space X is a homeomorphism from X to a simplicial complex.
Question (Poincar´e1899): Does every smooth manifold admit a triangulation?
Answer (Cairns, Whitehead 1940): Yes. Every smooth manifold has an essentially unique piecewise linear (PL) structure,∼ and therefore it is triangulable.
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 2 / 20 We can ask this about arbitrary triangulations, or about the more natural PL (combinatorial) triangulations.
Triangulations of manifolds
A triangulation of a topological space X is a homeomorphism from X to a simplicial complex.
Question (Poincar´e1899): Does every smooth manifold admit a triangulation?
Answer (Cairns, Whitehead 1940): Yes. Every smooth manifold has an essentially unique piecewise linear (PL) structure,∼ and therefore it is triangulable.
Question (Kneser 1924): Does every topological manifold admit a triangulation?
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 2 / 20 Triangulations of manifolds
A triangulation of a topological space X is a homeomorphism from X to a simplicial complex.
Question (Poincar´e1899): Does every smooth manifold admit a triangulation?
Answer (Cairns, Whitehead 1940): Yes. Every smooth manifold has an essentially unique piecewise linear (PL) structure,∼ and therefore it is triangulable.
Question (Kneser 1924): Does every topological manifold admit a triangulation? We can ask this about arbitrary triangulations, or about the more natural PL (combinatorial) triangulations.
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 2 / 20 PL triangulations are equivalent to PL structures on the manifold (up to isomorphism).
Triangulations of manifolds
A triangulation is called combinatorial (or PL) if the link of every vertex is (PL homeomorphic to) a sphere. Clearly, every space that admits a combinatorial triangulation is a manifold.
v →
S Lk(v)
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 3 / 20 Triangulations of manifolds
A triangulation is called combinatorial (or PL) if the link of every vertex is (PL homeomorphic to) a sphere. Clearly, every space that admits a combinatorial triangulation is a manifold.
v →
S Lk(v)
PL triangulations are equivalent to PL structures on the manifold (up to isomorphism).
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 3 / 20 Triangulations of manifolds
However, manifolds can have non-PL triangulations. Example: the double suspension of a n homology sphere M with π1(M) = 1. 6 .
M
.
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 4 / 20 Triangulations of manifolds
However, manifolds can have non-PL triangulations. Example: the double suspension of a n homology sphere M with π1(M) = 1. 6
Σ(M) M
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 5 / 20 We have Σ2M = Sn+2 (cf. Edwards, Cannon 1970s), but Lk(v) = ΣM is not a manifold. ∼ ∼
Triangulations of manifolds
However, manifolds can have non-PL triangulations. Example: the double suspension of a n homology sphere M with π1(M) = 1. 6
Σ2(M)
Σ(M)
v M
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 6 / 20 Triangulations of manifolds
However, manifolds can have non-PL triangulations. Example: the double suspension of a n homology sphere M with π1(M) = 1. 6
Σ2(M)
Σ(M)
v M
We have Σ2M = Sn+2 (cf. Edwards, Cannon 1970s), but Lk(v) = ΣM is not a manifold. ∼ ∼
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 6 / 20 dim 5 (cf. Sullivan, Casson, Kirby-Siebenmann 1960s) ≥ ∼ There exist non-PL manifolds 4 M is PL ∆(M) = 0 H (M; Z/2) ⇐⇒ ∈ 3 If they exist, PL structures are classified by elements in H (M; Z/2) (failure of the Hauptvermutung for manifolds)
dim 4 (cf. Donaldson, Freedman 1982): smooth (=PL) structures may not exist, or may not be unique ∼
PL triangulations
dim 3: unique smooth and PL structures ≤
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 7 / 20 dim 4 (cf. Donaldson, Freedman 1982): smooth (=PL) structures may not exist, or may not be unique ∼
PL triangulations
dim 3: unique smooth and PL structures ≤ dim 5 (cf. Sullivan, Casson, Kirby-Siebenmann 1960s) ≥ ∼ There exist non-PL manifolds 4 M is PL ∆(M) = 0 H (M; Z/2) ⇐⇒ ∈ 3 If they exist, PL structures are classified by elements in H (M; Z/2) (failure of the Hauptvermutung for manifolds)
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 7 / 20 PL triangulations
dim 3: unique smooth and PL structures ≤ dim 5 (cf. Sullivan, Casson, Kirby-Siebenmann 1960s) ≥ ∼ There exist non-PL manifolds 4 M is PL ∆(M) = 0 H (M; Z/2) ⇐⇒ ∈ 3 If they exist, PL structures are classified by elements in H (M; Z/2) (failure of the Hauptvermutung for manifolds)
dim 4 (cf. Donaldson, Freedman 1982): smooth (=PL) structures may not exist, or may not be unique ∼
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 7 / 20 dim 5: The idea is to look at the possible links of simplices of codimension n + 1. They are n-dimensional≥ homology spheres.
σ
Lk(σ)
We can rephrase triangulation questions in terms of the n-dimensional homology cobordism group.
Arbitrary triangulations
dim 4: (Casson) the Freedman E8 manifold is non-triangulable
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 8 / 20 Arbitrary triangulations
dim 4: (Casson) the Freedman E8 manifold is non-triangulable dim 5: The idea is to look at the possible links of simplices of codimension n + 1. They are n-dimensional≥ homology spheres.
σ
Lk(σ)
We can rephrase triangulation questions in terms of the n-dimensional homology cobordism group. Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 8 / 20 It turns out that Θn = 0 for n = 3, but Θ3 = 0. Z 6 Z 6
The homology cobordism group
n n n Θ = Y oriented, PL, H∗(Y ) = H∗(S ) / , Z { } ∼ n+1 Y0 Y1 compact, oriented, PL W with ∂W = ( Y0) Y1 and H∗(W , Yi ; Z) = 0. ∼ ⇐⇒ ∃ − ∪
4 Y0 W Y1
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 9 / 20 The homology cobordism group
n n n Θ = Y oriented, PL, H∗(Y ) = H∗(S ) / , Z { } ∼ n+1 Y0 Y1 compact, oriented, PL W with ∂W = ( Y0) Y1 and H∗(W , Yi ; Z) = 0. ∼ ⇐⇒ ∃ − ∪
4 Y0 W Y1
It turns out that Θn = 0 for n = 3, but Θ3 = 0. Z 6 Z 6
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 9 / 20 Rokhlin homomorphism µ :Θ3 /2, µ(Y ) = σ(W )/8 (mod 2) where W is any compact, Z Z smooth, spin 4-manifold with boundary→ Y . For example µ(S3) = 0, µ(Poincar´esphere) = 1. Hence Θ3 = 1. Z 6 It is known that Θ3 has a ∞ subgroup (Furuta, Fintushel-Stern 1990s), and at least a Z Z Z summand (cf. Frøyshov). ∼
∞ ∞ Unknown: Does it have torsion? A Z summand? Is it Z ?
The three-dimensional homology cobordism group
We can work with smooth (instead of PL) 4-diml. cobordisms.
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 10 / 20 It is known that Θ3 has a ∞ subgroup (Furuta, Fintushel-Stern 1990s), and at least a Z Z Z summand (cf. Frøyshov). ∼
∞ ∞ Unknown: Does it have torsion? A Z summand? Is it Z ?
The three-dimensional homology cobordism group
We can work with smooth (instead of PL) 4-diml. cobordisms. Rokhlin homomorphism µ :Θ3 /2, µ(Y ) = σ(W )/8 (mod 2) where W is any compact, Z Z smooth, spin 4-manifold with boundary→ Y . For example µ(S3) = 0, µ(Poincar´esphere) = 1. Hence Θ3 = 1. Z 6
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 10 / 20 ∞ ∞ Unknown: Does it have torsion? A Z summand? Is it Z ?
The three-dimensional homology cobordism group
We can work with smooth (instead of PL) 4-diml. cobordisms. Rokhlin homomorphism µ :Θ3 /2, µ(Y ) = σ(W )/8 (mod 2) where W is any compact, Z Z smooth, spin 4-manifold with boundary→ Y . For example µ(S3) = 0, µ(Poincar´esphere) = 1. Hence Θ3 = 1. Z 6 It is known that Θ3 has a ∞ subgroup (Furuta, Fintushel-Stern 1990s), and at least a Z Z Z summand (cf. Frøyshov). ∼
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 10 / 20 The three-dimensional homology cobordism group
We can work with smooth (instead of PL) 4-diml. cobordisms. Rokhlin homomorphism µ :Θ3 /2, µ(Y ) = σ(W )/8 (mod 2) where W is any compact, Z Z smooth, spin 4-manifold with boundary→ Y . For example µ(S3) = 0, µ(Poincar´esphere) = 1. Hence Θ3 = 1. Z 6 It is known that Θ3 has a ∞ subgroup (Furuta, Fintushel-Stern 1990s), and at least a Z Z Z summand (cf. Frøyshov). ∼
∞ ∞ Unknown: Does it have torsion? A Z summand? Is it Z ?
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 10 / 20 Mn (n 5) is triangulable an obstruction is zero in H5(M; ker(µ)). This could be replaced≥ with an equivalent⇐⇒ obstruction in H5(M; ), if we knew that Θ3 had no torsion Z Z with µ = 1. Triangulations (if they exist) are classified by elements in H4(M; ker(µ)).
Triangulations of manifolds in dim 5 ≥
By the work of Galewski-Stern and Matumoto in the 1970s:
There exist non-triangulable manifolds in dim 5 the exact sequence ≥ ⇐⇒ 3 µ 0 ker(µ) Θ Z/2 0 −→ −→ Z −→ −→ does not split. (M., 2013: It does not split; i.e. [Y ] Θ3 , 2[Y ] = 0, µ(Y ) = 1) 6 ∃ ∈ Z
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 11 / 20 Triangulations (if they exist) are classified by elements in H4(M; ker(µ)).
Triangulations of manifolds in dim 5 ≥
By the work of Galewski-Stern and Matumoto in the 1970s:
There exist non-triangulable manifolds in dim 5 the exact sequence ≥ ⇐⇒ 3 µ 0 ker(µ) Θ Z/2 0 −→ −→ Z −→ −→ does not split. (M., 2013: It does not split; i.e. [Y ] Θ3 , 2[Y ] = 0, µ(Y ) = 1) 6 ∃ ∈ Z Mn (n 5) is triangulable an obstruction is zero in H5(M; ker(µ)). This could be replaced≥ with an equivalent⇐⇒ obstruction in H5(M; ), if we knew that Θ3 had no torsion Z Z with µ = 1.
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 11 / 20 Triangulations of manifolds in dim 5 ≥
By the work of Galewski-Stern and Matumoto in the 1970s:
There exist non-triangulable manifolds in dim 5 the exact sequence ≥ ⇐⇒ 3 µ 0 ker(µ) Θ Z/2 0 −→ −→ Z −→ −→ does not split. (M., 2013: It does not split; i.e. [Y ] Θ3 , 2[Y ] = 0, µ(Y ) = 1) 6 ∃ ∈ Z Mn (n 5) is triangulable an obstruction is zero in H5(M; ker(µ)). This could be replaced≥ with an equivalent⇐⇒ obstruction in H5(M; ), if we knew that Θ3 had no torsion Z Z with µ = 1. Triangulations (if they exist) are classified by elements in H4(M; ker(µ)).
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 11 / 20 1 Yang-Mills theory (used by Fintushel-Stern and Furuta to show Θ3 is infinitely generated); Z 2 Seiberg-Witten theory 3 Heegaard Floer homology
In this talk we will discuss (2) and (3).
Methods for studying Θ3 Z
This is all about 4D smooth cobordisms, so we use gauge theory or symplectic geometry.
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 12 / 20 In this talk we will discuss (2) and (3).
Methods for studying Θ3 Z
This is all about 4D smooth cobordisms, so we use gauge theory or symplectic geometry.
1 Yang-Mills theory (used by Fintushel-Stern and Furuta to show Θ3 is infinitely generated); Z 2 Seiberg-Witten theory 3 Heegaard Floer homology
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 12 / 20 Methods for studying Θ3 Z
This is all about 4D smooth cobordisms, so we use gauge theory or symplectic geometry.
1 Yang-Mills theory (used by Fintushel-Stern and Furuta to show Θ3 is infinitely generated); Z 2 Seiberg-Witten theory 3 Heegaard Floer homology
In this talk we will discuss (2) and (3).
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 12 / 20 - in dim 3, we use them to obtain Seiberg-Witten Floer homology (cf. Kronheimer-Mrowka, Frøyshov, Marcolli-Wang, M.) and in fact a Seiberg-Witten Floer stable homotopy type (M.)
- In retrospect, all the homology cobordism invariants from SW theory can be obtained through the local equivalence group (cf. Stoffregen)
Seiberg-Witten equations
- system of nonlinear PDEs, elliptic (mod gauge), sensitive to the smooth structure (in dim.4)
- in the presence of a spin structure, they have a Pin(2) symmetry, where
1 1 Pin(2) = S jS C jC = H. ∪ ⊂ ⊕
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 13 / 20 - In retrospect, all the homology cobordism invariants from SW theory can be obtained through the local equivalence group (cf. Stoffregen)
Seiberg-Witten equations
- system of nonlinear PDEs, elliptic (mod gauge), sensitive to the smooth structure (in dim.4)
- in the presence of a spin structure, they have a Pin(2) symmetry, where
1 1 Pin(2) = S jS C jC = H. ∪ ⊂ ⊕
- in dim 3, we use them to obtain Seiberg-Witten Floer homology (cf. Kronheimer-Mrowka, Frøyshov, Marcolli-Wang, M.) and in fact a Seiberg-Witten Floer stable homotopy type (M.)
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 13 / 20 Seiberg-Witten equations
- system of nonlinear PDEs, elliptic (mod gauge), sensitive to the smooth structure (in dim.4)
- in the presence of a spin structure, they have a Pin(2) symmetry, where
1 1 Pin(2) = S jS C jC = H. ∪ ⊂ ⊕
- in dim 3, we use them to obtain Seiberg-Witten Floer homology (cf. Kronheimer-Mrowka, Frøyshov, Marcolli-Wang, M.) and in fact a Seiberg-Witten Floer stable homotopy type (M.)
- In retrospect, all the homology cobordism invariants from SW theory can be obtained through the local equivalence group (cf. Stoffregen)
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 13 / 20 = Pin(2)-equivariant spaces (suspension spectra) X with X S1 = sphere / local LE { } equivalence
1 X1 X2 f : X1 X2, g : X2 X1, Pin(2)-equivariant and equivalences on S -fixed point∼ sets⇐⇒ ∃ → →
This is modelled on the Seiberg-Witten maps induced by homology cobordisms. (chain local equivalence) is defined similarly, with chain complexes. CLE
The local equivalence group
There exist homomorphisms of Abelian groups
Θ3 Z → LE → CLE [Y ] [SWF(Y )] [C∗(SWF(Y ); F)], F = Z/2 → → where SWF(Y ) is the Seiberg-Witten Floer stable homotopy type, and
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 14 / 20 This is modelled on the Seiberg-Witten maps induced by homology cobordisms. (chain local equivalence) is defined similarly, with chain complexes. CLE
The local equivalence group
There exist homomorphisms of Abelian groups
Θ3 Z → LE → CLE [Y ] [SWF(Y )] [C∗(SWF(Y ); F)], F = Z/2 → → where SWF(Y ) is the Seiberg-Witten Floer stable homotopy type, and = Pin(2)-equivariant spaces (suspension spectra) X with X S1 = sphere / local LE { } equivalence
1 X1 X2 f : X1 X2, g : X2 X1, Pin(2)-equivariant and equivalences on S -fixed point∼ sets⇐⇒ ∃ → →
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 14 / 20 (chain local equivalence) is defined similarly, with chain complexes. CLE
The local equivalence group
There exist homomorphisms of Abelian groups
Θ3 Z → LE → CLE [Y ] [SWF(Y )] [C∗(SWF(Y ); F)], F = Z/2 → → where SWF(Y ) is the Seiberg-Witten Floer stable homotopy type, and = Pin(2)-equivariant spaces (suspension spectra) X with X S1 = sphere / local LE { } equivalence
1 X1 X2 f : X1 X2, g : X2 X1, Pin(2)-equivariant and equivalences on S -fixed point∼ sets⇐⇒ ∃ → →
This is modelled on the Seiberg-Witten maps induced by homology cobordisms.
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 14 / 20 The local equivalence group
There exist homomorphisms of Abelian groups
Θ3 Z → LE → CLE [Y ] [SWF(Y )] [C∗(SWF(Y ); F)], F = Z/2 → → where SWF(Y ) is the Seiberg-Witten Floer stable homotopy type, and = Pin(2)-equivariant spaces (suspension spectra) X with X S1 = sphere / local LE { } equivalence
1 X1 X2 f : X1 X2, g : X2 X1, Pin(2)-equivariant and equivalences on S -fixed point∼ sets⇐⇒ ∃ → →
This is modelled on the Seiberg-Witten maps induced by homology cobordisms. (chain local equivalence) is defined similarly, with chain complexes. CLE
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 14 / 20 ∗ In fact, H 1 (SWF(Y ); F) looks like F[U] (F[U]-torsion part). We let δ(Y ) be 1/2 the S ⊕ minimal grading in the F[U] tower. For the Poincare sphere we have δ(P) = 1. The existence of a surjective homomorphism 3 3 δ :Θ Z shows that Θ has a Z summand: Z → Z Θ3 = [P] ker(δ). Z ⊕
Numerical invariants
Prototype: The Frøyshov homomorphism
3 δ Θ Z, Z → LE → CLE −→ obtained from the S1-equivariant Seiberg-Witten Floer (co)homology of Y , which is a module over ∗ ∗ ∞ HS1 (pt) = H (CP ) = F[U], deg(U) = 2
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 15 / 20 For the Poincare sphere we have δ(P) = 1. The existence of a surjective homomorphism 3 3 δ :Θ Z shows that Θ has a Z summand: Z → Z Θ3 = [P] ker(δ). Z ⊕
Numerical invariants
Prototype: The Frøyshov homomorphism
3 δ Θ Z, Z → LE → CLE −→ obtained from the S1-equivariant Seiberg-Witten Floer (co)homology of Y , which is a module over ∗ ∗ ∞ HS1 (pt) = H (CP ) = F[U], deg(U) = 2
∗ In fact, H 1 (SWF(Y ); F) looks like F[U] (F[U]-torsion part). We let δ(Y ) be 1/2 the S ⊕ minimal grading in the F[U] tower.
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 15 / 20 Numerical invariants
Prototype: The Frøyshov homomorphism
3 δ Θ Z, Z → LE → CLE −→ obtained from the S1-equivariant Seiberg-Witten Floer (co)homology of Y , which is a module over ∗ ∗ ∞ HS1 (pt) = H (CP ) = F[U], deg(U) = 2
∗ In fact, H 1 (SWF(Y ); F) looks like F[U] (F[U]-torsion part). We let δ(Y ) be 1/2 the S ⊕ minimal grading in the F[U] tower. For the Poincare sphere we have δ(P) = 1. The existence of a surjective homomorphism 3 3 δ :Θ Z shows that Θ has a Z summand: Z → Z Θ3 = [P] ker(δ). Z ⊕
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 15 / 20 Corollary (M., 2013): There is no 2-torsion in Θ3 with µ = 1. (Hence, non-triangulable Z manifolds exist in dim 5.) ≥ Indeed: 2[Y ] = 0 Y Y β(Y ) = β( Y ) = β(Y ) β(Y ) = 0 µ(Y ) = 0. ⇒ ∼ − ⇒ − − ⇒ ⇒ Alternate construction of α, β, γ: F. Lin, 2014.
Pin(2) invariants
∗ Similarly to the Frøyshov invariant, we can use HPin(2)(SWF(Y ); F) to construct maps (not homomorphisms) 3 α,β,γ Θ / / / Z, Z LE CLE that satisfy α, β, γ(mod 2) = µ, α( Y ) = γ(Y ), β( Y ) = β(Y ). − − − −
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 16 / 20 Indeed: 2[Y ] = 0 Y Y β(Y ) = β( Y ) = β(Y ) β(Y ) = 0 µ(Y ) = 0. ⇒ ∼ − ⇒ − − ⇒ ⇒ Alternate construction of α, β, γ: F. Lin, 2014.
Pin(2) invariants
∗ Similarly to the Frøyshov invariant, we can use HPin(2)(SWF(Y ); F) to construct maps (not homomorphisms) 3 α,β,γ Θ / / / Z, Z LE CLE that satisfy α, β, γ(mod 2) = µ, α( Y ) = γ(Y ), β( Y ) = β(Y ). − − − − Corollary (M., 2013): There is no 2-torsion in Θ3 with µ = 1. (Hence, non-triangulable Z manifolds exist in dim 5.) ≥
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 16 / 20 Alternate construction of α, β, γ: F. Lin, 2014.
Pin(2) invariants
∗ Similarly to the Frøyshov invariant, we can use HPin(2)(SWF(Y ); F) to construct maps (not homomorphisms) 3 α,β,γ Θ / / / Z, Z LE CLE that satisfy α, β, γ(mod 2) = µ, α( Y ) = γ(Y ), β( Y ) = β(Y ). − − − − Corollary (M., 2013): There is no 2-torsion in Θ3 with µ = 1. (Hence, non-triangulable Z manifolds exist in dim 5.) ≥ Indeed: 2[Y ] = 0 Y Y β(Y ) = β( Y ) = β(Y ) β(Y ) = 0 µ(Y ) = 0. ⇒ ∼ − ⇒ − − ⇒ ⇒
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 16 / 20 Pin(2) invariants
∗ Similarly to the Frøyshov invariant, we can use HPin(2)(SWF(Y ); F) to construct maps (not homomorphisms) 3 α,β,γ Θ / / / Z, Z LE CLE that satisfy α, β, γ(mod 2) = µ, α( Y ) = γ(Y ), β( Y ) = β(Y ). − − − − Corollary (M., 2013): There is no 2-torsion in Θ3 with µ = 1. (Hence, non-triangulable Z manifolds exist in dim 5.) ≥ Indeed: 2[Y ] = 0 Y Y β(Y ) = β( Y ) = β(Y ) β(Y ) = 0 µ(Y ) = 0. ⇒ ∼ − ⇒ − − ⇒ ⇒ Alternate construction of α, β, γ: F. Lin, 2014.
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 16 / 20 Open problem: Describe the structure of , in general, and use it to understand more about Θ3 . LE CLE Z
More numerical invariants
Θ3 δ , Z / / / Z LE CLE α,β,γ κi ,κoi δ,δ¯ $ ZZZ
where δ,¯ δ come from Z/4-equivariant SWFH:
Z/4 = 1, 1, j, j Pin(2) = C jC { − − } ⊂ ⊕ and κi , i = 0, 1; κoi , i = 0,..., 7 come from Pin(2)-equivariant K-theory and KO-theory (M., J. Lin)
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 17 / 20 More numerical invariants
Θ3 δ , Z / / / Z LE CLE α,β,γ κi ,κoi δ,δ¯ $ ZZZ
where δ,¯ δ come from Z/4-equivariant SWFH:
Z/4 = 1, 1, j, j Pin(2) = C jC { − − } ⊂ ⊕ and κi , i = 0, 1; κoi , i = 0,..., 7 come from Pin(2)-equivariant K-theory and KO-theory (M., J. Lin) Open problem: Describe the structure of , in general, and use it to understand more about Θ3 . LE CLE Z Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 17 / 20 - replacement for SW theory:
+ S1 HF = HM = H∗ (SWF), HFc = HMg = H∗(SWF)
cf. (Kutluhan-Lee-Taubes, Taubes} + Colin-Ghiggini-Honda) + Lidman-M. - easier to compute, in fact algorithmically computable (cf. Sarkar-Wang, Lipshitz-Ozsv´ath-Thurston for HFc , M.-Ozsv´ath-Thurston for HF ±) One can recover the Frøyshov invariant δ using HF + (Ozsv´ath-Szab´o’s d-correction term).
Heegaard Floer homology
3 + − Ozsv´ath-Szab´o, 2001: Y HFc (Y ), HF (Y ), HF (Y ) using Lagrangian Floer homology on the symmetric product of a Heegaard surface for Y
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 18 / 20 - easier to compute, in fact algorithmically computable (cf. Sarkar-Wang, Lipshitz-Ozsv´ath-Thurston for HFc , M.-Ozsv´ath-Thurston for HF ±) One can recover the Frøyshov invariant δ using HF + (Ozsv´ath-Szab´o’s d-correction term).
Heegaard Floer homology
3 + − Ozsv´ath-Szab´o, 2001: Y HFc (Y ), HF (Y ), HF (Y ) using Lagrangian Floer homology on the symmetric product of a Heegaard surface for Y - replacement for SW theory:
+ S1 HF = HM = H∗ (SWF), HFc = HMg = H∗(SWF)
cf. (Kutluhan-Lee-Taubes, Taubes} + Colin-Ghiggini-Honda) + Lidman-M.
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 18 / 20 One can recover the Frøyshov invariant δ using HF + (Ozsv´ath-Szab´o’s d-correction term).
Heegaard Floer homology
3 + − Ozsv´ath-Szab´o, 2001: Y HFc (Y ), HF (Y ), HF (Y ) using Lagrangian Floer homology on the symmetric product of a Heegaard surface for Y - replacement for SW theory:
+ S1 HF = HM = H∗ (SWF), HFc = HMg = H∗(SWF)
cf. (Kutluhan-Lee-Taubes, Taubes} + Colin-Ghiggini-Honda) + Lidman-M. - easier to compute, in fact algorithmically computable (cf. Sarkar-Wang, Lipshitz-Ozsv´ath-Thurston for HFc , M.-Ozsv´ath-Thurston for HF ±)
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 18 / 20 Heegaard Floer homology
3 + − Ozsv´ath-Szab´o, 2001: Y HFc (Y ), HF (Y ), HF (Y ) using Lagrangian Floer homology on the symmetric product of a Heegaard surface for Y - replacement for SW theory:
+ S1 HF = HM = H∗ (SWF), HFc = HMg = H∗(SWF)
cf. (Kutluhan-Lee-Taubes, Taubes} + Colin-Ghiggini-Honda) + Lidman-M. - easier to compute, in fact algorithmically computable (cf. Sarkar-Wang, Lipshitz-Ozsv´ath-Thurston for HFc , M.-Ozsv´ath-Thurston for HF ±) One can recover the Frøyshov invariant δ using HF + (Ozsv´ath-Szab´o’s d-correction term).
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 18 / 20 + Z/4 Conjecture: HFI ∼= H∗ (SWF) (we do not know how to recover the whole Pin(2) symmetry yet)
It suffices to give invariants δ,¯ δ :Θ3 , computable for Seifert fibrations, large Z / Z surgeries on alternating knots, connected sums of these (Hendricks-M.-Zemke, Dai-M., Dai-Stoffregen 2016-7).
more constraints on which 3-manifolds are homology cobordant to each other. - new proofs that Θ3 has a ∞ subgroup (Stoffregen using Pin(2)-equiv. SW theory; Dai-M. Z Z using HFI )
Involutive Heegaard Floer homology
Hendricks, M., 2015: the conjugation symmetry on Heegaard Floer complexes + − HFId(Y ), HFI (Y ), HFI (Y )
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 19 / 20 It suffices to give invariants δ,¯ δ :Θ3 , computable for Seifert fibrations, large Z / Z surgeries on alternating knots, connected sums of these (Hendricks-M.-Zemke, Dai-M., Dai-Stoffregen 2016-7).
more constraints on which 3-manifolds are homology cobordant to each other. - new proofs that Θ3 has a ∞ subgroup (Stoffregen using Pin(2)-equiv. SW theory; Dai-M. Z Z using HFI )
Involutive Heegaard Floer homology
Hendricks, M., 2015: the conjugation symmetry on Heegaard Floer complexes + − HFId(Y ), HFI (Y ), HFI (Y ) + Z/4 Conjecture: HFI ∼= H∗ (SWF) (we do not know how to recover the whole Pin(2) symmetry yet)
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 19 / 20 more constraints on which 3-manifolds are homology cobordant to each other. - new proofs that Θ3 has a ∞ subgroup (Stoffregen using Pin(2)-equiv. SW theory; Dai-M. Z Z using HFI )
Involutive Heegaard Floer homology
Hendricks, M., 2015: the conjugation symmetry on Heegaard Floer complexes + − HFId(Y ), HFI (Y ), HFI (Y ) + Z/4 Conjecture: HFI ∼= H∗ (SWF) (we do not know how to recover the whole Pin(2) symmetry yet)
It suffices to give invariants δ,¯ δ :Θ3 , computable for Seifert fibrations, large Z / Z surgeries on alternating knots, connected sums of these (Hendricks-M.-Zemke, Dai-M., Dai-Stoffregen 2016-7).
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 19 / 20 - new proofs that Θ3 has a ∞ subgroup (Stoffregen using Pin(2)-equiv. SW theory; Dai-M. Z Z using HFI )
Involutive Heegaard Floer homology
Hendricks, M., 2015: the conjugation symmetry on Heegaard Floer complexes + − HFId(Y ), HFI (Y ), HFI (Y ) + Z/4 Conjecture: HFI ∼= H∗ (SWF) (we do not know how to recover the whole Pin(2) symmetry yet)
It suffices to give invariants δ,¯ δ :Θ3 , computable for Seifert fibrations, large Z / Z surgeries on alternating knots, connected sums of these (Hendricks-M.-Zemke, Dai-M., Dai-Stoffregen 2016-7).
more constraints on which 3-manifolds are homology cobordant to each other.
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 19 / 20 Involutive Heegaard Floer homology
Hendricks, M., 2015: the conjugation symmetry on Heegaard Floer complexes + − HFId(Y ), HFI (Y ), HFI (Y ) + Z/4 Conjecture: HFI ∼= H∗ (SWF) (we do not know how to recover the whole Pin(2) symmetry yet)
It suffices to give invariants δ,¯ δ :Θ3 , computable for Seifert fibrations, large Z / Z surgeries on alternating knots, connected sums of these (Hendricks-M.-Zemke, Dai-M., Dai-Stoffregen 2016-7).
more constraints on which 3-manifolds are homology cobordant to each other. - new proofs that Θ3 has a ∞ subgroup (Stoffregen using Pin(2)-equiv. SW theory; Dai-M. Z Z using HFI )
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 19 / 20 The maps δ, δ,¯ δ factor through I. Open question: What is I as an Abelian group? Can we use it to study Θ3 , e.g. to show it Z ∞ has a Z summand?
An analogue of CLE
There is a homomorphism
Θ3 I, [Y ] [CF −(Y ), the conjugation involution] Z → → 2 where I = free F[U]-complexes C∗ with automorphism ι, ι id, −1 { −1 ' U H∗(C) = F[U, U ] / ∼ } ∼ −1 C∗ D∗ f : C D, g : D C module homomorphisms that induce ∼= on U H∗, and∼ such that⇐⇒f ∃ι ιf ,→gι ιg. → ' '
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 20 / 20 Open question: What is I as an Abelian group? Can we use it to study Θ3 , e.g. to show it Z ∞ has a Z summand?
An analogue of CLE
There is a homomorphism
Θ3 I, [Y ] [CF −(Y ), the conjugation involution] Z → → 2 where I = free F[U]-complexes C∗ with automorphism ι, ι id, −1 { −1 ' U H∗(C) = F[U, U ] / ∼ } ∼ −1 C∗ D∗ f : C D, g : D C module homomorphisms that induce ∼= on U H∗, and∼ such that⇐⇒f ∃ι ιf ,→gι ιg. → ' ' The maps δ, δ,¯ δ factor through I.
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 20 / 20 An analogue of CLE
There is a homomorphism
Θ3 I, [Y ] [CF −(Y ), the conjugation involution] Z → → 2 where I = free F[U]-complexes C∗ with automorphism ι, ι id, −1 { −1 ' U H∗(C) = F[U, U ] / ∼ } ∼ −1 C∗ D∗ f : C D, g : D C module homomorphisms that induce ∼= on U H∗, and∼ such that⇐⇒f ∃ι ιf ,→gι ιg. → ' ' The maps δ, δ,¯ δ factor through I. Open question: What is I as an Abelian group? Can we use it to study Θ3 , e.g. to show it Z ∞ has a Z summand?
Ciprian Manolescu (UCLA) Homology cobordism and triangulations August 3, 2018 20 / 20