Torus Manifolds in Equivariant Complex Bordismmasuda/toric2014_osaka/Darby(slide).pdf · Torus Manifolds in Equivariant Complex Bordism ... Alastair Darby Torus Manifolds in Equivariant

Torus Manifolds in Equivariant Complex Bordism

Alastair Darby

Toric Topology 2014 in Osaka

January 24, 2014

Table of contents

1 Stably Complex Torus ManifoldsDefinitionsThe Tangential Representation

2 Oriented Torus GraphsDefinitions & ExamplesTorus PolynomialsBoundary Operator

3 Equivariant Complex BordismDefinitionsRestriction to Fixed Point Data

4 Equivariant K -theory Characteristic NumbersTheoremCorollaries

5 Omnioriented Quasitoric Manifolds

Stably Complex Torus ManifoldsOriented Torus Graphs

Equivariant Complex BordismEquivariant K -theory Characteristic Numbers

Omnioriented Quasitoric Manifolds

DefinitionsThe Tangential Representation

Stably Complex Torus Manifolds

Definition (Torus Manifold)

A torus manifold is a 2n-dimensional smooth compact manifold Mwith an effective smooth T n-action whose fixed point set isnon-empty.

Note that our fixed point set is finite; we only have isolated fixedpoints.

Definition (Stably Complex Torus Manifold)

A stably complex torus manifold is torus manifold with a complexT n-structure on

τ(M)⊕ R2k ,

for some large k .

Alastair Darby Torus Manifolds in Equivariant Complex Bordism





The Tangential Representation

Let M2n be a stably complex torus manifold and p ∈ MT n. We

have a complex T n-structure on

(τ(M)⊕ R2k)|p ∼= TpM ⊕ R2k ∼= τ(p)⊕ νMp ⊕ R2k .

So we can write

TpM = V1 ⊕ · · · ⊕ Vn, where Vi ∈ Hom(T n,S1) ∼= Zn.

Since the T n-action is effective we have:

Lemma

The irreducible T n-representations V1, . . . ,Vn form a basis ofHom(T n,S1) ∼= Zn.






The Tangential Representation

Notice TpM has two orientations:

one from its complex structure

one from the canonical orientation of M.

Definition (Sign of p)

For each isolated fixed point p of a stably complex torus manifold,the sign of p is given by

σ(p) :=

+1, if the two orientations coincide;

−1, if the two orientations differ.





Definitions & ExamplesTorus PolynomialsBoundary Operator

Torus Graphs (Maeda, Masuda & Panov)

Let

Γ be an n-valent connected graph with n ≥ 1.

V(Γ) denote the set of vertices.

E(Γ) denote the set of oriented edges.

i(e) t(e)e

For p ∈ V(Γ), define

E(Γ)p := e ∈ E(Γ) | i(e) = p.






Torus Graphs (Maeda, Masuda & Panov)

Definition (Torus Axial Function)

A torus axial function is a map

α : E(Γ) −→ Hom(T n,S1) ∼= Zn,

satisfying the following conditions:

1 α(e) = ±α(e);

2 elements of α(E(Γ)p) form a basis of Zn;

3 α(E(Γ)t(e)) ≡ α(E(Γ)i(e)) mod α(e), for any e ∈ E(Γ).

Definition (Torus Graph)

A torus graph is a pair (Γ, α) consisting of an n-valent graph Γwith a torus axial function α.


Example

Example (Torus Manifold)

Let M2n be a torus manifold. Define an n-valent graph ΓM where

V(ΓM) = MT n;

E(ΓM) = 2-dim submanifolds of M fixed by a T n−1 ≤ T n.

Every S ∈ E(ΓM) is diffeomorphic to a sphere and contains exactlytwo T n-fixed points. The summands TpM = V1(p)⊕ · · · ⊕ Vn(p)correspond to the edges E(ΓM)p. We assign each e ∈ E(ΓM)p toits corresponding Vi (p). This gives a function

αM : E(ΓM) −→ Hom(T n, S1),

which satisfies the three conditions of being a torus axial functionand we get a torus graph (ΓM , αM).





Oriented Torus Graphs

Definition

An orientation of a torus graph (Γ, α) is an assignment

σ : V(Γ) −→ ±1,

satisfying

σ(i(e))α(e) = −σ(i(e))α(e), for every e ∈ E(Γ).

Example (Stably Complex Torus Manifold)

Set σ(p), for p ∈ MT n= V(ΓM), to agree with the definition of the

sign of an isolated fixed point of a stably complex torus manifold.






Free Exterior Algebra

Definition

Let Jn denote the set of non-trivial elements of Hom(T n, S1) ∼= Zn.

Consider the free exterior Z-algebra on the set Jn:

Λ(Jn),

e.g. V ∧ V = 0 and V ∧W = −W ∧ V .

Definition (Faithful Polynomials)

We call an exterior polynomial in Λn(Jn) faithful if theindeterminates from each monomial form a basis of Zn.






Torus Polynomials

Suppose (Γ, α, σ) is an oriented torus graph. For a vertex p, orderthe basis elements α(E(Γ)p) = α(e1), . . . , α(en) so that

det[α(e1) · · ·α(en)] = σ(p).

This defines a faithful exterior monomial

µp = α(e1) ∧ · · · ∧ α(en) ∈ Λn(Jn), ∀p ∈ V(Γ).

Definition (Torus Polynomial)

The torus polynomial of an oriented torus graph (Γ, α, σ) is thefaithful exterior polynomial

g(Γ, α, σ) :=∑

p∈V(Γ)

µp ∈ Λn(Jn).






Definition

Definition

Define J∗n to be the set of non-trivial elements of Hom(S1,T n).

For each faithful exterior polynomial h ∈ Λn(Jn) we can obtain adual polynomial h∗ ∈ Λn(J∗n).We now define a chain complex (Λk(J∗n), dk) as follows: for eachmonomial s1 ∧ · · · ∧ sk ∈ Λk(J∗n), with all si ∈ J∗n ,

dk(s1∧· · ·∧sk) :=

∑ki=1(−1)i+1s1 ∧ · · · ∧ si ∧ · · · ∧ sk , if k > 1;

1 if k = 1.

and d0(1) = 0. It is easy to see that d2 = 0.






Theorem

Theorem (D.)

Let h ∈ Λn(Jn) be a faithful polynomial. Then h = g(Γ, α, σ) isthe torus polynomial of an oriented torus graph if and only ifd(h∗) = 0.

Let Kn denote the abelian group of all faithful exterior polynomialsh ∈ Λn(Jn) such that d(h∗) = 0.





DefinitionsRestriction to Fixed Point Data

Definitions

LetΩU:T n

m

denote the geometric equivariant complex bordism groups ofm-dimensional stably complex T n-manifolds. We have acommutative ring

ΩU:T n

∗ :=⊕m≥0

ΩU:T n

m

via the diagonal T n-action on the cartesian product of twoT n-manifolds.

Definition

Let ZU:T n

∗ ⊂ ΩU:T n

∗ denote the subring given by elements that canbe represented by a stably complex T n-manifold where theT n-action is effective.





DefinitionsRestriction to Fixed Point Data

Restriction to Fixed Point Data

We have a monomorphism by ‘restriction to fixed point data’:

ϕ : ZU:T n

∗ −→ Z[Jn]

[M] 7−→∑

p∈MTn

σ(p)m∏i=1

Vi (p),

where TpM = V1(p)⊕ · · · ⊕ Vm. When ∗ = 2n we obtain thecommutative diagram of abelian groups

ZU:T n

2n

g

||

ϕ

##Kn

f // Z[Jn]

where f (s1 ∧ · · · ∧ sn) = det[s1 · · · sn]s1 · · · sn for a faithfulmonomial s1 ∧ · · · ∧ sn ∈ Λn(Jn).





TheoremCorollaries

Equivariant K -theory Characteristic Numbers

Let S ⊂ K ∗(BT n+) denote the multiplicative subset generated by

the K -theory Euler classes λ−1(V ) =∑

i≥0(−1)iλi (V ) of thebundles ET n ×T n V → BT n, for V ∈ Jn.

Theorem (Hattori ’74)

There is a commutative pullback square with all maps injective

ZU:T n

∗Ψ //

ϕ

K ∗(BT n+)JtK

λ

Z[Jn]S−1Ψ // S−1K ∗(BT n

+)JtK

where t = (t1, t2, . . . ) is a sequence of indeterminates.





TheoremCorollaries

Equivariant K -theory Characteristic Numbers

The coefficients of Ψ[M] are known as equivariant K -theorycharacteristic numbers for M. Again, when ∗ = 2n we have

ZU:T n

2nΨ //

ϕ

g

||

K ∗(BT n+)JtK

λ

Kn

f ""Z[Jn]

S−1Ψ // S−1K ∗(BT n+)JtK





TheoremCorollaries

Theorem

ZU:T n

2nΨ //

ϕ

g

||

K ∗(BT n+)JtK

λ

Kn

f ""Z[Jn]

S−1Ψ // S−1K ∗(BT n+)JtK

Theorem (D.)

Every polynomial h ∈ Kn satisfies

(S−1Ψ f )(h) ∈ λ(K ∗(BT n+)JtK).





TheoremCorollaries

Corollaries

Corollary

We have an isomorphism of abelian groups

ZU:T n

2n∼= Kn.

Define the graded rings

Ξ∗ :=⊕n≥0

ZU:T n

2n∼= K∗ :=

⊕n≥0

Kn.

Warning

These are non-commutative rings.





TheoremCorollaries

Corollaries

Suppose M2n is a non-bounding stably complex torus manifold.Then g [M] ∈ Kn is a non-zero faithful polynomial in Λn(Jn) suchthat d(g [M]∗) = 0. Any such exterior polynomial must have atleast n + 1 monomials.

Corollary

As a strict lower bound, n + 1 is the minimum number of fixedpoints of a non-bounding stably complex torus manifold.





Definition

A quasitoric manifold is an even-dimensional smooth closedmanifold M2n with a locally standard smooth T n-action such thatthe orbit space is a simple polytope P.

Definition

A quasitoric pair (P, λ) consists of a combinatorial oriented simplen-polytope P and a map

λ : F(P) −→ Hom(S1,T n) ∼= Zn

that satisfies:

λ(Fi1), . . . , λ(Fin) forms a basis of Hom(S1,T n) whenever (?)

Fi1 ∩ · · · ∩ Fin is a vertex of P.





Quasitoric Pairs

We have a bijection

Quasitoric manifolds with a stably complex T n-structure

l

Quasitoric pairs

We define a product on quasitoric pairs

(P1, λ1)× (P2, λ2) := (P1 × P2, λ1 × λ2),

where the characteristic map is defined as

(λ1 × λ2)(Fi × P2) = (λ1(Fi ), 0, . . . , 0) and

(λ1 × λ2)(P1 × F ′i ) = (0, . . . , 0, λ2(F ′i )).





Ring of Quasitoric Pairs

Definition (Ring of Quasitoric Pairs)

Denote the free abelian group generated by all quasitoric pairs byQ∗, where we may interpret + as disjoint union and grade Q∗ bythe dimension of the polytope.

The multiplication depends on the ordering of P1 × P2 so Q∗forms a graded non-commutative ring. We have a homomorphismof non-commutative graded rings

M : Q∗ −→ Ξ∗,

by constructing the omnioriented quasitoric manifold associated toa quasitoric pair.





Conjecture

Conjecture

The homomorphism M : Q∗ −→ Ξ∗ is surjective, that is, everyclass is Ξ∗ contains an omnioriented quasitoric manifold.

True for n = 1, 2.





Thank you!


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Torus Manifolds in Equivariant Complex Bordismmasuda/toric2014_osaka/Darby(slide).pdf · Torus Manifolds in Equivariant Complex Bordism ... Alastair Darby Torus Manifolds in Equivariant