Cyclic p-fold power construction

Construct an equivariant singular diagonal. [F2, F3, F7, F8, A1] For the identity simplex ${\iota }_n:{\mathrm{\Delta }}^n\to {\mathrm{\Delta }}^n$, construct elements $\mathrm{\Phi }(e_j\otimes {\iota }_n)\in C_{\ast }({\mathrm{\Delta }}^n;{\mathbb{F}}_p{)}^{\otimes p}$ by induction on $j+n$. The already defined boundary is a cycle because the cyclic-resolution differential squares to zero. The affine contraction of ${\mathrm{\Delta }}^n$ to its first vertex, [F8], and the iterated chain equivalence [F7] make its augmented $p$fold tensor complex acyclic, so a filling exists. [A1] selects one filling for each nonempty extension problem. Define the other $C_p$-translates equivariantly, fix degree zero to be the iterated Alexander--Whitney diagonal, and put

Define the singular coefficients and prove well-definedness. [F2, F3, step 1.1] For a degree-$q$ cocycle $c$, define

Prove additivity and identify $D_0$. [F3, F4, F7, step 2.1] For cocycles $c,d$, the mixed words in $(c+d{)}^{\otimes p}-c^{\otimes p}-d^{\otimes p}$ form free $C_p$-orbits. Taking the lexicographically least word in each finite orbit writes their sum as ${\mathrm{Tr}}_1^{C_p}z$. The explicit contraction of $W$ after forgetting its action, together with [F7], makes restriction from equivariant to ordinary cohomology onto. Hence [F4]'s $\mathrm{Tr}\mathrm{Res}=p=0$ shows that diagonal pullback kills the mixed class. Uniqueness of the $[w_j]$ coordinates gives $D_j(x+y)=D_j(x)+D_j(y)$.

Compare with finite cellular coefficients. [F1, F2, F6, F7, F8, A1, step 1.1, step 2.1] For a finite regular $K$, barycentric subdivision of each closed cell defines a cell-carried chain map $\iota :C_{\ast }^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(K;{\mathbb{F}}_p)\to C_{\ast }^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(K;{\mathbb{F}}_p)$. By [F6] it is a homology isomorphism. Its mapping cone is acyclic; [A1] chooses complements to its boundary subspaces, whose inverse boundary maps contract the cone. Dualizing proves that ${\iota }^{\ast }$ is a cohomology isomorphism.

Establish the sharp range on finite regular complexes. [F1, step 3.2] Restriction to the $q$-skeleton is injective on $H^q$ and an isomorphism in lower degrees by the cellular cochain complex. Collapse its $(q-1)$-skeleton and map each $q$-cell to $S^q$ with the integer degree representing the chosen coefficient of a cellular cocycle. This gives a map $K^q\to S^q$ pulling the sphere generator back to the class. Naturality therefore reduces $D_j$ for $j>(p-1)q$ to a class in $H^{pq-j}(S^q;{\mathbb{F}}_p)$ below degree $q$. It is zero except possibly in degree zero. In that last case $j=pq$ and $q>0$; restriction to a point sends the sphere generator to zero, so additivity sends $D_{pq}$ to zero, while $H^0(S^q)\to H^0(\ast )$ is injective. Thus $D_j=0$ throughout the stated range.

Apply the normalizer action at odd primes. [F3, F13, step 3.2] For $a\in {\mathbb{F}}_p^{\times }$, multiplication by $a$ permutes the $p$ tensor positions and conjugates $T$ to $T^a$. Its sign is computed from the Vandermonde product:

Derive the external product formula. [F1, F3, F7, step 1.1, step 3.2] Take the tensor product of the two equivariant power cocycles and pull it back along the diagonal $C_p\to C_p\times C_p$. The shuffle moving $p$ degree-$s$ factors past the degree-$r$ factors contributes $(-1{)}^{p(p-1)rs\mathrm{/}2}$. The cyclic diagonal in [F3] gives every split with coefficient one when $p=2$; for odd $p$, its even coordinate has only the even--even splits, since the odd--odd coefficient $p(p-1)\mathrm{/}2$ is zero in ${\mathbb{F}}_p$. Comparing the unique $w_k$ coordinates yields the two displayed external formulas. The carrier comparison in step 1.1 makes this chain calculation valid for arbitrary spaces, not only finite complexes.

Relate adjacent coefficients by the positive Bockstein. [F3, F4, F10, F11, F12, F13, step 2.1, step 3.1] Lift $c$ by its canonical residues [F11]. Writing $\delta \tilde{c}=ph$, the coboundary of ${\tilde{c}}^{\otimes p}$ divided by $p$ is the cyclic sum of the $p$ words with one $h$ and $p-1$ copies of $c$. For odd $p$ this is a transfer, so step 3.1's transfer argument makes the Bockstein of the pulled back total class zero. By [F3] and [F10], our positive convention has $\beta w_{2r}=0$ and $\beta w_{2r-1}=w_{2r}$. Applying the signed derivation rule [F12] to ${\sum }_j{w}_j\times D_j(x)$ and comparing even and odd coordinates gives

Compute the circle coefficient. [F1, F3, step 3.2] Give $S^1$ two oriented edges $J_1,J_2$ with common boundary and let the cocycle $z$ take values $1,0$ on them. Steenrod--Epstein's carried map is obtained recursively from the interval contraction. At resolution degree $p-1=2m$, its displayed finite sum has the single surviving multi-index ${\alpha }_i={\beta }_i=0$ and hence

Compute every top coefficient. [F9, step 4.1, step 4.3, step 4.5] For $q\ge 1$, let $u$ be the generator in degree $q-1$ on $S^{q-1}$ and let $z$ be the circle class. [F9] makes $u\times z$ nonzero. The sharp range already proved leaves only the two top factors in step 4.3, so

Pass the finite identities to every space and check boundaries. [F5, A1, step 2.1, step 3.1, step 3.2, step 4.1, step 4.2, step 4.3, step 4.4, step 5.1] For fixed $p,q,j$, each residual in steps 4.1, 4.2, and 5.1 is a natural map between fixed singular cohomology degrees. It vanishes on finite regular complexes by steps 3.2--5.1, so [F5] makes it vanish on every space. The external and Bockstein formulas were already proved directly on singular cochains.