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Borel and Cartan equivariant cohomology for a compact torus

Definition

For a nonnegative integer $r$ , let $T = (R / Z)^{r}$ be a compact torus with its standard product Lie-group smooth structure and period-one basis, and let $M$ be a smooth manifold without boundary, not assumed compact, with a smooth left action $a : T \times M \to M$ . Work over $R$ and use the chosen product Milnor model $E T = (E_{S^{1}})^{r}$ and $B T = E T / T$ from the preceding coefficient lemma. Give $E T \times M$ the right diagonal action $(e, p) \cdot t = (e \cdot t, t^{- 1} \cdot p)$ and define $E T \times_{T} M : = (E T \times M) / T .$ The Borel equivariant cohomology is the graded ring $H_{T}^{*} (M; R) : = H_{s i n g}^{*} (E T \times_{T} M; R) .$ For $M = {*}$ this gives $H_{T}^{*} (*; R) = H^{*} (B T; R)$ ; the selected degree-two normalization in ${Sym}_{R} (t^{*})$ is the isomorphism established in the preceding coefficient lemma.

The quotient map $q : E T \times M \to E T \times_{T} M$ and projection $π ([e, p]) = [e]$ give the associated-space square $\begin{array}{ccc} E T \times M & \overset{q}{\to} & E T \times_{T} M \\ ↓ {pr}_{1} & ↓ π \\ E T & \overset{q_{T}}{\to} & B T \end{array}$ Its TikZ source is

\begin{tikzcd}
ET\times M \arrow[r,"q"] \arrow[d,"\operatorname{pr}_1"'] & ET\times_T M \arrow[d,"\pi"]\\
ET \arrow[r,"q_T"'] & BT
\end{tikzcd}

where $q_{T} : E T \to B T$ is the orbit quotient. The projection $π$ is well-defined because $[e \cdot t] = [e]$ for every $t \in T$ . The square commutes because both composites send $(e, p)$ to $[e]$ .

Let $t = T_{e} T$ , with basis $e_{1}, \dots, e_{r}$ and dual basis $u_{1}, \dots, u_{r}$ . Regard ${Sym}_{R} (t^{*})$ as a polynomial algebra by assigning each linear generator degree two. Use the trivial $T$ -action on this polynomial factor, and the induced left action $t \cdot α = (a_{t^{- 1}})^{*} α$ on forms. It is a left action because pullback reverses composition: $a_{t^{- 1}}^{*} a_{s^{- 1}}^{*} = (a_{s^{- 1}} \circ a_{t^{- 1}})^{*} = a_{(t s)^{- 1}}^{*}$ . The Cartan complex is the graded-commutative algebra $C_{T} (M) : = ({Sym}_{R} (t^{*}) \otimes_{R} Ω^{*} (M))^{T},$ with multiplication $(P \otimes α) (Q \otimes β) = P Q \otimes (α \land β)$ , where $\land$ is the differential-form wedge product and all tensor products are algebraic. The superscript denotes the fixed subspace for the diagonal action, and a polynomial of degree $j$ tensored with a $k$ -form has total degree $2 j + k$ . For $ξ \in t$ , use the library fundamental vector field $X_{ξ} (p) = {\frac{d}{d s} ∣}_{s = 0} \exp_{T} (- s ξ) \cdot p .$ On polynomial-valued forms define the degree-one Cartan operator $d_{T} (P \otimes α) = P \otimes d α - \sum_{i = 1}^{r} (u_{i} P) \otimes ι_{X_{e_{i}}} α,$ extended linearly. The tensor $\sum_{i} u_{i} \otimes X_{e_{i}}$ is basis-independent. The Cartan equivariant cohomology in this item means $H^{*} (C_{T} (M), d_{T})$ ; the proof below establishes $d_{T}^{2} = 0$ on $C_{T} (M)$ .

This definition does not assert a general isomorphism between the Borel and Cartan cohomology rings. It defines neither equivariant Chow groups nor equivariant quotient-stack Borel–Moore homology, and no rational-coefficient comparison is being used in the real Cartan complex.

Definition references

[Smooth manifold structure] Smooth manifolds and their smooth charts

[Jointly smooth left action] Smooth left actions of Lie groups

[Standard product torus and period-one basis] The standard product torus as a compact Lie group

[Negative-exponential fundamental-field convention] Fundamental vector fields for a left action

[Smooth differential forms] A smooth differential $k$ -form

[Symmetric algebra] Symmetric algebra of a vector space

[Differential-form wedge product] The wedge product of differential forms

[Graded-commutative algebra of differential forms] Differential forms form a graded commutative algebra

[Singular cohomology and its cup product] Singular cohomology with coefficients

[Singular cochain complex] Singular cochain complex with coefficients

[Singular cup product] Singular cup product on cochains

[Singular cohomology ring structure] Cup product is natural, unital and associative

[Exterior derivative] The exterior derivative by the invariant vector-field formula

[Interior product] Interior product of a form by a vector field

[Lie derivative] The Lie derivative of a differential form

[Graded-derivation rule for the exterior derivative] The exterior derivative is a graded derivation

[Graded-antiderivation rule for contraction] Interior product is a graded antiderivation

[Naturality of the exterior derivative under pullback] The exterior derivative commutes with pullback

[Cartan formula] Cartan's magic formula

Facts & Assumptions

Given: AC, a nonnegative integer $r$ , the fixed product torus presentation, the smooth manifold $M$ , and its smooth left $T$ -action.

[A1]

Assume the full Axiom of Choice, as required by the coefficient calculation. The Axiom of Choice

[F1]

For the selected product Milnor model, $H^{*} (B T; R) ≅ {Sym}_{R} (t^{*})$ with the stated degree-two normalization. Real cohomology of the chosen torus classifying space

[F2]

A smooth left action is a jointly smooth map satisfying the left action law, so each $a_{t}$ is a diffeomorphism. Smooth left actions of Lie groups

[F3]

The fundamental field is $X_{ξ} (p) = {\frac{d}{d s} ∣}_{0} \exp_{T} (- s ξ) \cdot p$ , and $ξ \mapsto X_{ξ}$ is linear as part of the stated Lie-algebra-homomorphism convention. Fundamental vector fields for a left action

[F4]

A smooth differential $k$ -form is a smooth section of $⋀^{k} T^{*} M$ ; the exterior derivative maps $Ω^{k} (M)$ to $Ω^{k + 1} (M)$ . A smooth differential $k$ -form The exterior derivative by the invariant vector-field formula

[F5]

Singular cohomology with real coefficients is the cocycle quotient of the real singular cochain complex. Singular cohomology with coefficients

[F6]

The singular cup product is defined by the front/back-face cochain formula. Singular cup product on cochains

[F7]

Singular cohomology with this cup product is a unital graded ring. Cup product is natural, unital and associative

[F8]

The exterior derivative satisfies $d^{2} = 0$ . The exterior derivative squares to zero

[F9]

The interior product is pointwise insertion of the vector field into the first slot of an alternating form; hence two insertions anticommute. Interior product of a form by a vector field

[F10]

The Lie derivative of a form is defined by differentiating pullback along the local flow. The Lie derivative of a differential form The Lie derivative of a tensor field

[F11]

The exterior derivative is a degree-one graded derivation of the algebra of forms. The exterior derivative is a graded derivation

[F12]

Contraction by a vector field is a degree-minus-one graded antiderivation. Interior product is a graded antiderivation

[F13]

For every smooth map $F$ , $d (F^{*} ω) = F^{*} (d ω)$ . The exterior derivative commutes with pullback

[F14]

For every vector field $X$ and form $α$ , $L_{X} α = d (ι_{X} α) + ι_{X} (d α)$ . Cartan's magic formula

[F15]

The symmetric algebra is the commutative graded quotient of the tensor algebra by $v \otimes w - w \otimes v$ . Symmetric algebra of a vector space

[F16]

Differential forms form an associative graded-commutative algebra under the pointwise wedge product. The wedge product of differential forms Differential forms form a graded commutative algebra

[F17]

The countable-choice hypothesis ${A C}_{ω}$ requires an indexed selector for every sequence of nonempty sets. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F18]

The fixed product torus is a compact connected abelian Lie group, and each period-one basis vector exponentiates to its coordinate circle subgroup as specified in the local torus definition. The standard product torus as a compact Lie group

[F19]

For the empty space there are no singular simplices, so all singular cochain groups and its singular cohomology vanish. Singular cochain complex with coefficients

Proof

1.1A1F1F5F6F7given

The right diagonal action is a group action by homeomorphisms, so its orbit quotient $E T \times_{T} M$ is a topological space. By [F5]–[F7] its real singular cohomology has the stated cup-product ring structure; at $M = {*}$ the quotient is $B T$ , whose coefficient ring is identified by [F1].

1.2F2F3F4F9F11F12F15F16algebra

Pullback by each $a_{t^{- 1}}$ preserves wedge products: evaluating on tangent vectors shows that pulling back the pointwise alternating product gives the product of the pullbacks. Thus the diagonal $T$ -action is by algebra automorphisms and its fixed subspace is closed under the multiplication defined by the algebraic tensor-product rule and [F15]–[F16]. Since each polynomial generator has degree two, $d$ raises form degree by one, and contraction lowers form degree by one, both summands in $d_{T}$ raise total degree by one. By [F11] and [F12], each summand is a degree-one derivation because each $u_{i}$ has even degree. Under a change of basis of $t$ , the polynomial basis and vector-field basis transform contragrediently; linearity in [F3] then makes $\sum_{i} u_{i} \otimes X_{e_{i}}$ basis-independent, so $d_{T}$ is unchanged.

1.3A1F3F17construct

Let $(A_{n})_{n \in N}$ be any sequence of nonempty sets and let $F = {A_{n} : n \in N}$ . By [A1], choose a function $g$ on $F$ with $g (S) \in S$ for every $S \in F$ . Then $c (n) : = g (A_{n})$ is a function on $N$ , including when different indices have equal sets, and $c (n) \in A_{n}$ for every $n$ . This is exactly the indexed selector in [F17], so the countable-choice hypothesis of [F3] is satisfied.

2.1F2F3F9F13F18step 1.2step 1.3

Pullback naturality [F13] shows that $d$ preserves the $T$ -fixed forms. For $t \in T$ , the left action law and commutativity give $a_{t} \circ a_{\exp (- s e_{i})} = a_{\exp (- s e_{i})} \circ a_{t}$ ; differentiating at $s = 0$ shows $(a_{t})_{*} X_{e_{i}} = X_{e_{i}}$ . By the pointwise definition [F9], contraction by each $X_{e_{i}}$ preserves invariant forms, while $u_{i}$ is fixed by the stipulated trivial action on the polynomial factor. Both terms therefore preserve the diagonal $T$ -fixed subspace, so $d_{T} : C_{T} (M) \to C_{T} (M)$ is well-defined.

2.2F8F9F14step 1.2algebra

On polynomial-valued forms, expand the square as $d_{T}^{2} = d^{2} - \sum_{i} u_{i} (d ι_{X_{e_{i}}} + ι_{X_{e_{i}}} d) + \sum_{i, j} u_{i} u_{j} ι_{X_{e_{i}}} ι_{X_{e_{j}}} .$ The first term is zero by [F8]. The last sum is zero because $u_{i} u_{j} = u_{j} u_{i}$ while insertions into alternating forms anticommute by [F9]. The middle sum is $- \sum_{i} u_{i} L_{X_{e_{i}}}$ by [F14]. Hence $d_{T}^{2} α = - \sum_{i} u_{i} L_{X_{e_{i}}} α$ for every $α \in C_{T} (M)$ .

3.1F3F10F18step 2.1step 2.2

Put $ϕ_{s} = a_{\exp (- s e_{i})}$ . The local torus definition [F18] identifies $\exp (- s e_{i})$ with the coordinate one-parameter subgroup, so the group law gives $ϕ_{s + h} = ϕ_{s} \circ ϕ_{h}$ . Differentiating at $h = 0$ gives $\frac{d}{d s} ϕ_{s} (p) = (d ϕ_{s})_{p} X_{e_{i}} (p) = X_{e_{i}} (ϕ_{s} (p))$ by step 2.1. Thus $ϕ_{s}$ is the flow of $X_{e_{i}}$ . For an invariant form $α$ , $ϕ_{s}^{*} α = α$ for every $s$ ; differentiating at zero and using [F10] gives $L_{X_{e_{i}}} α = 0$ . Substitution in step 2.2 proves $d_{T}^{2} α = 0$ on $C_{T} (M)$ .

4.1F2F3F5F8F19step 3.1∎

If $r = 0$ , the sum in $d_{T}$ is empty and $d_{T} = d$ , whose square is zero by [F8]. If the action is trivial, every $X_{e_{i}} = 0$ and again $d_{T} = d$ on invariant forms. If $M = \emptyset$ , the form complex is the zero complex and the Borel construction is the empty space, whose singular cohomology is zero by [F19]. No compactness of $M$ , symplectic form, or Hamiltonian hypothesis was used.

Borel and Cartan equivariant cohomology for a compact torus

Definition

Definition references

Facts & Assumptions

Proof

Depends on

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