Equivariant p-fold external power and diagonal decomposition

Proof technique: construct the tensor power on cellular chains, compare choices by equivariant acyclic carriers, decompose diagonal cochains coordinatewise, and kill mixed terms by transfer.

$d(x_1\otimes \cdots \otimes x_p)={\sum }_{i=1}^p(-1{)}^{\mathrm{\mid }{x}_1\mathrm{\mid }+\cdots +\mathrm{\mid }{x}_{i-1}\mathrm{\mid }}{x}_1\otimes \cdots \otimes d{x}_i\otimes \cdots \otimes x_p\mathrm{.}$

This follows cell by cell from the oriented boundary of a product disk. Let $C_p=⟨T⟩$ rotate the tensor factors with the Koszul sign. Write $H_{C_p}^{\ast }(W\times K^p;{\mathbb{F}}_p)$ for the cohomology of ${\mathrm{Hom}}_{{\mathbb{F}}_p[C_p]}(W\otimes C^{\otimes p},{\mathbb{F}}_p)$.

Taking the whole target as carrier proves that any two free acyclic $\mathrm{\Gamma }$-resolutions admit augmentation-preserving comparison maps, unique up to equivariant chain homotopy. Taking $E=I\otimes W$ and target $I^p\otimes W$, with the endpoint maps $0\otimes w\mapsto 0^p\otimes w$ and $1\otimes w\mapsto 1^p\otimes w$, gives an equivariant map $h$ joining those ends. This is the only use of AC in the construction.

$\mathcal{P}(c)(w\otimes x_1\otimes \cdots \otimes x_p)=\epsilon (w)c(x_1)\cdots c(x_p)\mathrm{.}$

The tensor differential in step 1.1 and $cd=0$ show directly that $\delta \mathcal{P}(c)=0$. Rotating $p$ degree-$q$ inputs has sign $(-1{)}^{q^2(p-1)}$. This is $1$ for odd $p$, while for $p=2$ every sign is $1$ in ${\mathbb{F}}_2$; hence $\mathcal{P}(c)$ is $C_p$-equivariant. On the fiber selected by an augmented zero-cell $e_0$ of $W$, $\epsilon (e_0)=1$, so its restriction is exactly the cellular external cochain $c^{\otimes p}$ and represents $u\times \cdots \times u$.

$I^p\otimes W\otimes C^{\otimes p}⟶W\otimes (I\otimes C{)}^{\otimes p},$

and $\epsilon \otimes D^{\otimes p}$. The result is an equivariant cochain homotopy from $\mathcal{P}(c)$ to $\mathcal{P}(c^{\mathrm{\prime }})$, so their classes agree.

For another free acyclic resolution $V$, an augmentation-preserving comparison $W\to V$ from step 1.2 pulls the defining cochain on $V$ back literally to the defining cochain on $W$. Two comparison maps induce the same cohomology map because their equivariant chain homotopy gives the usual cochain coboundary. Comparisons in both directions have composites homotopic to the identities by the same uniqueness argument, so these maps are isomorphisms and the class is resolution-independent in the asserted sense.

$\mathcal{P}(c\circ f_{\mathrm{\#}})=\mathcal{P}(c)\circ (1_W\otimes f_{\mathrm{\#}}^{\otimes p})\mathrm{.}$

Subdivision maps and their composites are covered by the same comparison uniqueness, so the induced cohomology map is independent of all subdivisions and approximations. The equality proves naturality, while step 3.1 makes it independent of the chosen cocycle.

${\mathrm{Hom}}_{{\mathbb{F}}_p[C_p]}((W\otimes C{)}_n,{\mathbb{F}}_p)={⨁}_{j=0}^n{w}_j\otimes C^{n-j}\mathrm{.}$

The $W$-part of the cochain differential is zero, as computed in [F1], and the remaining coordinate differential is $(-1{)}^j{\delta }_C$. Therefore taking cycles and boundaries coordinatewise gives, without a splitting choice,

$H_{C_p}^n(W\times K;{\mathbb{F}}_p)={⨁}_{j=0}^n[w_j]\times H_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^{n-j}(K;{\mathbb{F}}_p)\mathrm{.}$

Pulling $\mathcal{P}(u)$ back along the equivariant map $1_W\times d$ and taking its unique coordinates defines the stated $D_j(u)$. A cellular approximation to $1_W\times d$ is equivalently a $C_p$-equivariant chain map ${\mathrm{\Phi }}_C:W\otimes C\to C^{\otimes p}$ carried by the cellwise diagonal. Existence and independence up to a carried equivariant homotopy follow from step 1.2. Evaluating the defining cocycle $\epsilon \otimes c^{\otimes p}$ after this approximation shows that its $w_j$ coordinate is exactly the cochain $z\mapsto c^{\otimes p}{\mathrm{\Phi }}_C(e_j\otimes z)$. Step 4.1 and uniqueness of the fixed basis coordinates prove naturality of every $D_j$.

${\mathrm{Tr}}_1^{C_p}(z)=(c+d{)}^{\otimes p}-c^{\otimes p}-d^{\otimes p}\mathrm{.}$

After tensoring with the invariant augmentation $\epsilon$, the difference $\mathcal{P}(c+d)-\mathcal{P}(c)-\mathcal{P}(d)$ is therefore the transfer of $\epsilon \otimes z$.

It remains to justify vanishing after diagonal pullback. Forgetting the $C_p$-action on the standard $W$, write an element of $R={\mathbb{F}}_p[s]\mathrm{/}(s^p)$ as $\sum a_i{s}^i$. Define

$\alpha \text{\hspace{0.17em}⁣}(\sum a_i{s}^i)={\sum }_{i=1}^{p-1}{a}_i{s}^{i-1},\lambda \text{\hspace{0.17em}⁣}(\sum a_i{s}^i)=a_{p-1}\mathrm{.}$

Use $\alpha$ as the contracting map from every even resolution degree to the next odd degree and $x\mapsto \lambda (x)\cdot 1$ from every odd degree to the next even degree. The identities $s\alpha (x)+\lambda (s^{p-1}x)\cdot 1=x$ in positive even degrees, $s^{p-1}\lambda (x)+\alpha (sx)=x$ in odd degrees, and $s\alpha (x)+\epsilon (x)\cdot 1=x$ in degree zero give an explicit contraction of $W$ to ${\mathbb{F}}_p$. Tensoring it with $C$ shows that ordinary cohomology of $W\times K$ is pulled back from $K$. Every such class is the restriction of the equivariant class $[w_0]\times a$ from step 5.1, so restriction from equivariant to ordinary cohomology is onto.

Transfer commutes with the diagonal pullback: for an equivariant chain map $f$ and an ordinary cochain $a$, direct substitution in the coset sum gives $\mathrm{Tr}(a\circ f)=\mathrm{Tr}(a)\circ f$. Given an ordinary class on $W\times K$, lift it through that onto restriction and apply [F2]; transfer of the class is zero because transfer after restriction is multiplication by $p=0$. Hence diagonal pullback kills the transferred mixed class above. Step 5.1's unique coordinate decomposition now gives $D_j(u+v)=D_j(u)+D_j(v)$ for every $j$.