GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-Sheaves and Equivariant Cohomology
Tom Baird - Memorial University of Newfoundland

25 March 2021

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

Outline1

GKM theory

2

The Chang-Skjelbred Theorem

3

Our main result

4

GKM-hypergraphs

5

GKM-sheaves

GKM-hypergraphs

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main resultLet
T = (S1 )r a compact torus.

GKM-hypergraphs

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphsLet
T = (S1 )r a compact torus.
X a finite T -CW complex (e.g. a compact, smooth
T -manifold).

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheavesLet
T = (S1 )r a compact torus.
X a finite T -CW complex (e.g. a compact, smooth
T -manifold).
HT∗ (X ) := H ∗ (ET ×T X ; C) the equivariant cohomology ring.

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheavesLet
T = (S1 )r a compact torus.
X a finite T -CW complex (e.g. a compact, smooth
T -manifold).
HT∗ (X ) := H ∗ (ET ×T X ; C) the equivariant cohomology ring.
A := HT∗ (point).

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheavesLet
T = (S1 )r a compact torus.
X a finite T -CW complex (e.g. a compact, smooth
T -manifold).
HT∗ (X ) := H ∗ (ET ×T X ; C) the equivariant cohomology ring.
A := HT∗ (point).
GKM theory is a collection of techniques for calculating HT∗ (X )and related invariants.

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

For a large class of T -manifolds with isolated fixed points, Goresky,
Kottwitz and MacPherson showed that HT∗ (X ) can be encoded
combinatorially in a finite graph: the GKM graph.

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

For a large class of T -manifolds with isolated fixed points, Goresky,
Kottwitz and MacPherson showed that HT∗ (X ) can be encoded
combinatorially in a finite graph: the GKM graph. The vertices
correspond to fixed points of the action

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

For a large class of T -manifolds with isolated fixed points, Goresky,
Kottwitz and MacPherson showed that HT∗ (X ) can be encoded
combinatorially in a finite graph: the GKM graph. The vertices
correspond to fixed points of the action and the edges are labelled
by non-trivial characters α ∈ Hom(T , S1 ).

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

For a large class of T -manifolds with isolated fixed points, Goresky,
Kottwitz and MacPherson showed that HT∗ (X ) can be encoded
combinatorially in a finite graph: the GKM graph. The vertices
correspond to fixed points of the action and the edges are labelled
by non-trivial characters α ∈ Hom(T , S1 ).
GKM-theory subsequently developed in several directions:

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

For a large class of T -manifolds with isolated fixed points, Goresky,
Kottwitz and MacPherson showed that HT∗ (X ) can be encoded
combinatorially in a finite graph: the GKM graph. The vertices
correspond to fixed points of the action and the edges are labelled
by non-trivial characters α ∈ Hom(T , S1 ).
GKM-theory subsequently developed in several directions:combinatorially by Guillemin and Zara,

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

For a large class of T -manifolds with isolated fixed points, Goresky,
Kottwitz and MacPherson showed that HT∗ (X ) can be encoded
combinatorially in a finite graph: the GKM graph. The vertices
correspond to fixed points of the action and the edges are labelled
by non-trivial characters α ∈ Hom(T , S1 ).
GKM-theory subsequently developed in several directions:combinatorially by Guillemin and Zara,
to a broader range of spaces by Guillemin and Holm,GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

For a large class of T -manifolds with isolated fixed points, Goresky,
Kottwitz and MacPherson showed that HT∗ (X ) can be encoded
combinatorially in a finite graph: the GKM graph. The vertices
correspond to fixed points of the action and the edges are labelled
by non-trivial characters α ∈ Hom(T , S1 ).
GKM-theory subsequently developed in several directions:combinatorially by Guillemin and Zara,
to a broader range of spaces by Guillemin and Holm,to equivariant intersection cohomology by Braden and
MacPherson who introduced the notion of Γ-sheaves on a
GKM-graph.

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

For a large class of T -manifolds with isolated fixed points, Goresky,
Kottwitz and MacPherson showed that HT∗ (X ) can be encoded
combinatorially in a finite graph: the GKM graph. The vertices
correspond to fixed points of the action and the edges are labelled
by non-trivial characters α ∈ Hom(T , S1 ).
GKM-theory subsequently developed in several directions:combinatorially by Guillemin and Zara,
to a broader range of spaces by Guillemin and Holm,to equivariant intersection cohomology by Braden and
MacPherson who introduced the notion of Γ-sheaves on a
GKM-graph.
GKM-sheaves provide a unified framework for these constructions.

GKM theory

The Chang-Skjelbred Theorem

Our main result

Given a T -space X , consider the filtration

GKM-hypergraphs

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

Given a T -space X , consider the filtration
X T = X0 ⊆ X1 ⊆ ... ⊆ Xr = X
where Xk is the union of orbits of dimension ≤ k.

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

Given a T -space X , consider the filtration
X T = X0 ⊆ X1 ⊆ ... ⊆ Xr = X
where Xk is the union of orbits of dimension ≤ k.
Theorem (The Chang-Skjelbred Theorem)
If HT∗ (X ) is a free A-module, then there is a natural exact sequencei∗δ

0 → HT∗ (X ) −→ HT∗ (X0 ) →− HT∗+1 (X1 , X0 )(1)

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

Given a T -space X , consider the filtration
X T = X0 ⊆ X1 ⊆ ... ⊆ Xr = X
where Xk is the union of orbits of dimension ≤ k.
Theorem (The Chang-Skjelbred Theorem)
If HT∗ (X ) is a free A-module, then there is a natural exact sequencei∗δ

0 → HT∗ (X ) −→ HT∗ (X0 ) →− HT∗+1 (X1 , X0 )where i ∗ is induced by inclusion X0 ⊆ X(1)GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

Given a T -space X , consider the filtration
X T = X0 ⊆ X1 ⊆ ... ⊆ Xr = X
where Xk is the union of orbits of dimension ≤ k.
Theorem (The Chang-Skjelbred Theorem)
If HT∗ (X ) is a free A-module, then there is a natural exact sequencei∗δ

0 → HT∗ (X ) −→ HT∗ (X0 ) →− HT∗+1 (X1 , X0 )(1)

where i ∗ is induced by inclusion X0 ⊆ X and δ is the coboundary
map for the pair (X1 , X0 ).

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

Given a T -space X , consider the filtration
X T = X0 ⊆ X1 ⊆ ... ⊆ Xr = X
where Xk is the union of orbits of dimension ≤ k.
Theorem (The Chang-Skjelbred Theorem)
If HT∗ (X ) is a free A-module, then there is a natural exact sequencei∗δ

0 → HT∗ (X ) −→ HT∗ (X0 ) →− HT∗+1 (X1 , X0 )(1)

where i ∗ is induced by inclusion X0 ⊆ X and δ is the coboundary
map for the pair (X1 , X0 ).
By work of Allday-Franz-Puppe, (1) is exact if and only if HT∗ (X )is a 2-syzygy,

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

Given a T -space X , consider the filtration
X T = X0 ⊆ X1 ⊆ ... ⊆ Xr = X
where Xk is the union of orbits of dimension ≤ k.
Theorem (The Chang-Skjelbred Theorem)
If HT∗ (X ) is a free A-module, then there is a natural exact sequencei∗δ

0 → HT∗ (X ) −→ HT∗ (X0 ) →− HT∗+1 (X1 , X0 )(1)

where i ∗ is induced by inclusion X0 ⊆ X and δ is the coboundary
map for the pair (X1 , X0 ).
By work of Allday-Franz-Puppe, (1) is exact if and only if HT∗ (X )is a 2-syzygy, meaning that there exists an exact sequence
0 → F0 → F1 → HT∗ (X )where F0 , F1 are free A-modules.

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

To a finite T -CW complex X , we associate a certain sheaf of
A-modules FX , which packages cohomological and combinatorial
information about (X1 , X0 ).

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

To a finite T -CW complex X , we associate a certain sheaf of
A-modules FX , which packages cohomological and combinatorial
information about (X1 , X0 ).
Theorem (Al-Jabea, B)
If HT∗ (X ) is a 2-syzygy, then there is a natural isomorphism
H 0 (FX ) ∼= HT∗ (X ).

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

To a finite T -CW complex X , we associate a certain sheaf of
A-modules FX , which packages cohomological and combinatorial
information about (X1 , X0 ).
Theorem (Al-Jabea, B)
If HT∗ (X ) is a 2-syzygy, then there is a natural isomorphism
H 0 (FX ) ∼= HT∗ (X ).
Furthermore, there is a natural exact sequenceδ

0 → H 0 (FX ) → HT∗ (X0 ) →− HT∗+1 (X1 , X0 ) → H 1 (FX ) → 0

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

To a finite T -CW complex X , we associate a certain sheaf of
A-modules FX , which packages cohomological and combinatorial
information about (X1 , X0 ).
Theorem (Al-Jabea, B)
If HT∗ (X ) is a 2-syzygy, then there is a natural isomorphism
H 0 (FX ) ∼= HT∗ (X ).
Furthermore, there is a natural exact sequenceδ

0 → H 0 (FX ) → HT∗ (X0 ) →− HT∗+1 (X1 , X0 ) → H 1 (FX ) → 0
and H n (FX ) = 0 for n ≥ 2.

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

To a finite T -CW complex X , we associate a certain sheaf of
A-modules FX , which packages cohomological and combinatorial
information about (X1 , X0 ).
Theorem (Al-Jabea, B)
If HT∗ (X ) is a 2-syzygy, then there is a natural isomorphism
H 0 (FX ) ∼= HT∗ (X ).
Furthermore, there is a natural exact sequenceδ

0 → H 0 (FX ) → HT∗ (X0 ) →− HT∗+1 (X1 , X0 ) → H 1 (FX ) → 0
and H n (FX ) = 0 for n ≥ 2.
The sheaf FX is called a GKM-sheaf and is defined on a
GKM-hypergraph ΓX .

GKM theory

The Chang-Skjelbred Theorem

Our main result

T ∼= (S1 )r a compact torus.

GKM-hypergraphs

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

T ∼= (S1 )r a compact torus.
Λ := Hom(T , U(1)) ⊂ t∗ the weight lattice.

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

T ∼= (S1 )r a compact torus.
Λ := Hom(T , U(1)) ⊂ t∗ the weight lattice.
P(Λ) := (Λ − {0})/scalars is the set of projective weights.

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

T ∼= (S1 )r a compact torus.
Λ := Hom(T , U(1)) ⊂ t∗ the weight lattice.
P(Λ) := (Λ − {0})/scalars is the set of projective weights.
α ∈ P(Λ) ↔ ker (α) ∈ {codimension one subtori of T }GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

T ∼= (S1 )r a compact torus.
Λ := Hom(T , U(1)) ⊂ t∗ the weight lattice.
P(Λ) := (Λ − {0})/scalars is the set of projective weights.
α ∈ P(Λ) ↔ ker (α) ∈ {codimension one subtori of T }A GKM-hypergraph Γ = (V, ∼) consists of

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

T ∼= (S1 )r a compact torus.
Λ := Hom(T , U(1)) ⊂ t∗ the weight lattice.
P(Λ) := (Λ − {0})/scalars is the set of projective weights.
α ∈ P(Λ) ↔ ker (α) ∈ {codimension one subtori of T }A GKM-hypergraph Γ = (V, ∼) consists of1

A finite set V called the vertices

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

T ∼= (S1 )r a compact torus.
Λ := Hom(T , U(1)) ⊂ t∗ the weight lattice.
P(Λ) := (Λ − {0})/scalars is the set of projective weights.
α ∈ P(Λ) ↔ ker (α) ∈ {codimension one subtori of T }A GKM-hypergraph Γ = (V, ∼) consists of1

A finite set V called the vertices

2

An equivalence relation ∼α on V for every α ∈ P(Λ).

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

A GKM-hypergraph Γ = (V, ∼) consists of,
A finite set V called the vertices
An equivalence relation ∼α on V for every α ∈ P(Λ).

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

A GKM-hypergraph Γ = (V, ∼) consists of,
A finite set V called the vertices
An equivalence relation ∼α on V for every α ∈ P(Λ).
A morphism of GKM-hypergraphs
φ : (V, ∼) 7→ (V 0 , ∼0 )GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

A GKM-hypergraph Γ = (V, ∼) consists of,
A finite set V called the vertices
An equivalence relation ∼α on V for every α ∈ P(Λ).
A morphism of GKM-hypergraphs
φ : (V, ∼) 7→ (V 0 , ∼0 )is a map φ : V → V 0 such that

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

A GKM-hypergraph Γ = (V, ∼) consists of,
A finite set V called the vertices
An equivalence relation ∼α on V for every α ∈ P(Λ).
A morphism of GKM-hypergraphs
φ : (V, ∼) 7→ (V 0 , ∼0 )is a map φ : V → V 0 such thatv ∼α w ⇒ φ(v ) ∼0α φ(w )for all α ∈ P(Λ).

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

To any finite T -CW complex X , we associate a GKM-hypergraphΓX = (VX , ∼) by

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

To any finite T -CW complex X , we associate a GKM-hypergraphΓX = (VX , ∼) by
VX := the set of path components of X T = π0 (X T )GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

To any finite T -CW complex X , we associate a GKM-hypergraphΓX = (VX , ∼) by
VX := the set of path components of X T = π0 (X T )For α ∈ P(Λ) and V , W ∈ VX ,V ∼α W ⇔ they lie in the same component of X ker(α) .

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

To any finite T -CW complex X , we associate a GKM-hypergraphΓX = (VX , ∼) by
VX := the set of path components of X T = π0 (X T )For α ∈ P(Λ) and V , W ∈ VX ,V ∼α W ⇔ they lie in the same component of X ker(α) .
This determines a functor
Γ : finite T -CW complexes 7→ GKM-hypergraphs

GKM theory

The Chang-Skjelbred Theorem

Our main result

The set of hyperedges of Γ = (V, ∼) is

GKM-hypergraphs

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

The set of hyperedges of Γ = (V, ∼) is
E := {(α, S) ∈ P(Λ) × ℘(V)| S is an ∼α equiv. class}

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

The set of hyperedges of Γ = (V, ∼) is
E := {(α, S) ∈ P(Λ) × ℘(V)| S is an ∼α equiv. class}
We have forgetful maps,
α : E → P(Λ)(the axial function),

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

The set of hyperedges of Γ = (V, ∼) is
E := {(α, S) ∈ P(Λ) × ℘(V)| S is an ∼α equiv. class}
We have forgetful maps,
α : E → P(Λ)(the axial function),

I : E → ℘(V)

(the incidence map).

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

The set of hyperedges of Γ = (V, ∼) is
E := {(α, S) ∈ P(Λ) × ℘(V)| S is an ∼α equiv. class}
We have forgetful maps,
α : E → P(Λ)(the axial function),

I : E → ℘(V)

(the incidence map).

Define the topological space Top(Γ) with underlying set V ∪ E

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

The set of hyperedges of Γ = (V, ∼) is
E := {(α, S) ∈ P(Λ) × ℘(V)| S is an ∼α equiv. class}
We have forgetful maps,
α : E → P(Λ)(the axial function),

I : E → ℘(V)

(the incidence map).

Define the topological space Top(Γ) with underlying set V ∪ E andbasic open sets
Uv := {v }Ue := {e} ∪ I (e)GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

The set of hyperedges of Γ = (V, ∼) is
E := {(α, S) ∈ P(Λ) × ℘(V)| S is an ∼α equiv. class}
We have forgetful maps,
α : E → P(Λ)(the axial function),

I : E → ℘(V)

(the incidence map).

Define the topological space Top(Γ) with underlying set V ∪ E andbasic open sets
Uv := {v }Ue := {e} ∪ I (e)GKM-morphisms induce continuous maps in this topology.

GKM theory

The Chang-Skjelbred Theorem

DenoteA := S(t∗C )Our main result

GKM-hypergraphs

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

DenoteA := S(t∗C ) ∼= HT∗ (point; C)

GKM-hypergraphs

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

DenoteA := S(t∗C ) ∼= HT∗ (point; C) ∼= C[u1 , ..., ur ]with deg(ui ) = 2.

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

DenoteA := S(t∗C ) ∼= HT∗ (point; C) ∼= C[u1 , ..., ur ]with deg(ui ) = 2.
Definition
A GKM-sheaf F over Γ is a sheaf of finitely generated, Z-graded
A-modules over Top(Γ), such that

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

DenoteA := S(t∗C ) ∼= HT∗ (point; C) ∼= C[u1 , ..., ur ]with deg(ui ) = 2.
Definition
A GKM-sheaf F over Γ is a sheaf of finitely generated, Z-graded
A-modules over Top(Γ), such that1

F is locally free over A .

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

DenoteA := S(t∗C ) ∼= HT∗ (point; C) ∼= C[u1 , ..., ur ]with deg(ui ) = 2.
Definition
A GKM-sheaf F over Γ is a sheaf of finitely generated, Z-graded
A-modules over Top(Γ), such that1

F is locally free over A .

2

The restriction F(Ue ) → F(I (e)) is an isomorphism modulo
α(e)−1 for all e ∈ E.

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

DenoteA := S(t∗C ) ∼= HT∗ (point; C) ∼= C[u1 , ..., ur ]with deg(ui ) = 2.
Definition
A GKM-sheaf F over Γ is a sheaf of finitely generated, Z-graded
A-modules over Top(Γ), such that1

F is locally free over A .

2

The restriction F(Ue ) → F(I (e)) is an isomorphism modulo
α(e)−1 for all e ∈ E.
F(Ue ) ∼= F(I (e)) for all but finitely many e ∈ E.

3

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

Denote by GKM(Γ) the category of GKM-sheaves on Γ.

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

Denote by GKM(Γ) the category of GKM-sheaves on Γ.
Pushforwards: Given φ : Γ1 → Γ2 , define a functorφ∗ : GKM(Γ1 ) 7→ GKM(Γ2 ),F → φ∗ (F)GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

Denote by GKM(Γ) the category of GKM-sheaves on Γ.
Pushforwards: Given φ : Γ1 → Γ2 , define a functorφ∗ : GKM(Γ1 ) 7→ GKM(Γ2 ),F → φ∗ (F)Invariants: If G acts by automorphisms on Γ, then may consider
the category of equivariant GKM-sheaves GKMG (Γ).

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

Denote by GKM(Γ) the category of GKM-sheaves on Γ.
Pushforwards: Given φ : Γ1 → Γ2 , define a functorφ∗ : GKM(Γ1 ) 7→ GKM(Γ2 ),F → φ∗ (F)Invariants: If G acts by automorphisms on Γ, then may consider
the category of equivariant GKM-sheaves GKMG (Γ).We obtainfunctorIvt : GKMG (Γ) 7→ GKM(Γ/G ),by pushing forward and taking invariants.

F 7→ (φ∗ (F))GGKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

Denote by GKM(Γ) the category of GKM-sheaves on Γ.
Pushforwards: Given φ : Γ1 → Γ2 , define a functorφ∗ : GKM(Γ1 ) 7→ GKM(Γ2 ),F → φ∗ (F)Invariants: If G acts by automorphisms on Γ, then may consider
the category of equivariant GKM-sheaves GKMG (Γ).We obtainfunctorIvt : GKMG (Γ) 7→ GKM(Γ/G ),F 7→ (φ∗ (F))G

by pushing forward and taking invariants.
Products: Define an external tensor product : GKM(Γ1 ) × GKM(Γ2 ) 7→ GKM(Γ1 × Γ2 )

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

Denote by GKM(Γ) the category of GKM-sheaves on Γ.
Pushforwards: Given φ : Γ1 → Γ2 , define a functorφ∗ : GKM(Γ1 ) 7→ GKM(Γ2 ),F → φ∗ (F)Invariants: If G acts by automorphisms on Γ, then may consider
the category of equivariant GKM-sheaves GKMG (Γ).We obtainfunctorIvt : GKMG (Γ) 7→ GKM(Γ/G ),F 7→ (φ∗ (F))G

by pushing forward and taking invariants.
Products: Define an external tensor product : GKM(Γ1 ) × GKM(Γ2 ) 7→ GKM(Γ1 × Γ2 )
F1  F2 := π1∗ (F1 ) ⊗ π2∗ (F2 )where πi : Γ1 × Γ2 → Γi is the projection morphism.

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

For a finite T -CW complex X , we construct a GKM-sheaf FX over
ΓX ,

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

For a finite T -CW complex X , we construct a GKM-sheaf FX over
ΓX , with stalks:
FX (UV ) := HT∗ (V ) ∼= H ∗ (V ) ⊗C A

where we identify vertices V with connected components of X T

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

For a finite T -CW complex X , we construct a GKM-sheaf FX over
ΓX , with stalks:
FX (UV ) := HT∗ (V ) ∼= H ∗ (V ) ⊗C A
andFX (UE ) := HT∗ (E )/TorA (HT∗ (E ))where we identify vertices V with connected components of X T
and hyperedges E with connected components of X ker(αE ) .

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

For a finite T -CW complex X , we construct a GKM-sheaf FX over
ΓX , with stalks:
FX (UV ) := HT∗ (V ) ∼= H ∗ (V ) ⊗C A
andFX (UE ) := HT∗ (E )/TorA (HT∗ (E ))where we identify vertices V with connected components of X T
and hyperedges E with connected components of X ker(αE ) .
Theorem
There is a canonical morphism of graded A-algebrasHT∗ (X ) → H 0 (FX )which is an isomorphism X if and only if HT∗ (X ) is 2-syzygy.

GKM theory

The Chang-Skjelbred Theorem

Idea of the proof:

Our main result

GKM-hypergraphs

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

Idea of the proof:
By Chang-Skjelbred and Allday-Franz-Puppe, we have an exact
sequencei∗δ

0 → HT∗ (X ) −→ HT∗ (X0 ) →− HT∗+1 (X1 , X0 )GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

Idea of the proof:
By Chang-Skjelbred and Allday-Franz-Puppe, we have an exact
sequencei∗δ

0 → HT∗ (X ) −→ HT∗ (X0 ) →− HT∗+1 (X1 , X0 )By construction V ⊂ Top(ΓX ) is an open set andFX (V) = HT∗ (X T ).

GKM-sheaves

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

Idea of the proof:
By Chang-Skjelbred and Allday-Franz-Puppe, we have an exact
sequencei∗δ

0 → HT∗ (X ) −→ HT∗ (X0 ) →− HT∗+1 (X1 , X0 )By construction V ⊂ Top(ΓX ) is an open set andFX (V) = HT∗ (X T ).
We show that a section in FM (V) extends to Top(Γ) if and only if
it lies in the kernel of δ.

GKM theory

The Chang-Skjelbred Theorem

Our main result

GKM-hypergraphs

GKM-sheaves

Al-Jabea, Ibrahem, and Thomas John Baird. ”Cohomology of
GKM-sheaves.” arXiv preprint arXiv:1806.01761 (2018).
C. Allday, M. Franz, V. Puppe, Equivariant cohomology,
syzygies and orbit structure, Trans. Amer. Math. Soc.
Volume 366, Number 12, December 2014, Pages 6567-6589.
T. Baird, GKM-sheaves and nonorientable surface grouprepresentations, Journal of Symplectic Geometry, Volume 12,
Number 4 (2014), 867-921.
T. Braden and R. MacPherson, From moment graphs tointersection cohomology, Mathematische Annalen 321 (2001),
no. 3, 533551.
M. Goresky, R. Kottwitz, and R. MacPherson, Equivariant
cohomology, Koszul duality, and the localization theorem,
Inventiones Mathematicae 131 (1997), no. 1, 2583.
V. Guillemin, T. S. Holm. ”GKM theory for torus actions with
nonisolated fixed points.” International Mathematics Research
Notices 2004.40: 2105-2124, 2004
V. Guillemin and C. Zara, 1-skeleta, Betti numbers, andGKM theory

The Chang-Skjelbred Theorem

Thank you!

Our main result

GKM-hypergraphs

GKM-sheaves