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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
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
DefinitionsThe Tangential Representation
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.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
DefinitionsThe Tangential Representation
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.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
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.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
Definitions & ExamplesTorus PolynomialsBoundary Operator
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.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
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).
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
Definitions & ExamplesTorus PolynomialsBoundary Operator
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.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
Definitions & ExamplesTorus PolynomialsBoundary Operator
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.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
Definitions & ExamplesTorus PolynomialsBoundary Operator
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).
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
Definitions & ExamplesTorus PolynomialsBoundary Operator
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.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
Definitions & ExamplesTorus PolynomialsBoundary Operator
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.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
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.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
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).
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
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.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
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
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
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).
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
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.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
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.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
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.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
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 )).
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
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.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
Conjecture
Conjecture
The homomorphism M : Q∗ −→ Ξ∗ is surjective, that is, everyclass is Ξ∗ contains an omnioriented quasitoric manifold.
True for n = 1, 2.
Alastair Darby Torus Manifolds in Equivariant Complex Bordism
Stably Complex Torus ManifoldsOriented Torus Graphs
Equivariant Complex BordismEquivariant K -theory Characteristic Numbers
Omnioriented Quasitoric Manifolds
Thank you!
Alastair Darby Torus Manifolds in Equivariant Complex Bordism