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Finite-cellular cyclic squares, Cartan formula, and cyclic-basis action

Statement

Assume AC and specialize the finite cellular cyclic-power construction to $p = 2$ . For a finite oriented regular cell complex $K$ and $x \in H_{c e l l}^{q} (K; F_{2})$ , write

$d^{*} P (x) = \sum_{r = 0}^{2 q} [w_{r}] \times D_{r} (x)$

and define

$S q_{c y c}^{i} (x) = D_{q - i} (x) (0 \leq i \leq q), S q_{c y c}^{i} (x) = 0 (i < 0 or i > q) .$

These operations are additive and natural,

$S q_{c y c}^{0} x = x, S q_{c y c}^{q} x = x ⌣ x,$

and they satisfy the external and internal Cartan formulas

$S q_{c y c}^{k} (x \times y) = \sum_{i + j = k} S q_{c y c}^{i} (x) \times S q_{c y c}^{j} (y),$

$S q_{c y c}^{k} (x ⌣ y) = \sum_{i + j = k} S q_{c y c}^{i} (x) ⌣ S q_{c y c}^{j} (y) .$

If $t = [w_{1}]$ is the degree-one generator of $H^{*} (B C_{2}; F_{2}) = F_{2} [t]$ , “computed on a finite skeleton” means computed on the finite regular simplicial model $Q_{N}$ constructed in step 5.1, not on the nonregular one-cell projective CW skeleton. For every $N > r + j$ , restriction identifies $t^{d}$ with a cellular class $t_{N}^{d}$ for $d \leq r + j$ , and

$S q_{c y c}^{j} (t^{r}) = (\binom{r}{j}) t^{r + j} .$

This finite-cellular lemma does not identify $S q_{c y c}$ with the singular cup- $i$ squares.

Facts & Assumptions

Given: AC, finite oriented regular cell complexes, the mod-two cyclic resolution $W$ , and the coefficient operations $D_{r}$ of the cyclic power.

[F1]

The finite cellular cyclic-power class is natural on finite regular complexes (Equivariant p-fold external power and diagonal decomposition).

[F2]

Its diagonal pullback has unique coefficients $D_{r}$ (Equivariant p-fold external power and diagonal decomposition).

[F3]

Its restriction to a zero-cell fiber is the ordinary external power (Equivariant p-fold external power and diagonal decomposition).

[F4]

For $p = 2$ , the cyclic-resolution quotient has $H^{*} (B C_{2}; F_{2}) = F_{2} [t]$ with $t = [w_{1}]$ (Free cyclic resolution, group cohomology, and cochain transfer).

[F5]

At $p = 2$ , the explicit cyclic-resolution diagonal on an even cell has every even--even and odd--odd split with coefficient one (Free cyclic resolution, group cohomology, and cochain transfer).

[F6]

AC supplies a choice function for every set-indexed family of nonempty sets (The Axiom of Choice).

[F7]

The external power is independent of its cocycle and free-resolution choices (Equivariant p-fold external power and diagonal decomposition).

[F8]

The explicit cyclic-resolution diagonal on an odd cell has every split with coefficient one (Free cyclic resolution, group cohomology, and cochain transfer).

[F9]

Every cyclic-power coefficient $D_{r}$ is additive (Equivariant p-fold external power and diagonal decomposition).

[F10]

The vertices of $sd L$ are the nonempty faces of $L$ , and its simplices are their strict chains (Barycentric subdivision of an abstract simplicial complex).

[F11]

Barycentric subdivision realizes homeomorphically, compatibly with subcomplex inclusions (Barycentric subdivision realizes homeomorphically).

[F12]

The usual projective CW structure has one cell in every degree through $N$ , with integral boundary $2$ in positive even degree and $0$ in odd degree (Real projective space cellular homology and the pinch map).

[F13]

Cellular homology computes singular homology naturally for cellular maps (Cellular homology computes singular homology), and a cellular map induces the corresponding cellular chain map (Cellular maps induce cellular chain maps).

[F14]

Under AC, evaluation identifies cohomology over a field naturally with the full dual of homology (Cohomology over a field is dual to homology over that field).

[F15]

Singular cup product is the Alexander--Whitney diagonal evaluation (Singular cup product on cochains), and Alexander--Whitney and shuffle are augmentation-preserving natural chain-homotopy inverses (Alexander--Whitney and shuffle are natural chain-homotopy inverses).

[F16]

Pullback in singular cohomology is a unital ring homomorphism (Cup product is natural, unital and associative).

Proof

Proof technique: compare powers across the cyclic-resolution diagonal, determine the zero-square scalar on spheres, and apply the resulting total square to the polynomial generator.

1.1

Define the finite-cellular operations. [given, F1, F2, F9] The formula in the statement merely reindexes the unique coefficients from [F2]. Additivity and naturality of every $S q_{c y c}^{i}$ follow from those of $D_{q - i}$ ; the two outside-range clauses are definitions. All operations here remain on the finite oriented regular cellular model.

1.2

Prove the external coefficient formula. [F2, F5, F7, F8] Let $x \in H_{c e l l}^{q} (K; F_{2})$ and $y \in H_{c e l l}^{s} (L; F_{2})$ . Regrouping the four factors shows on pure tensors that the external square of $x \times y$ is the product of the external squares of $x$ and $y$ . Compare the one cyclic resolution on the left with two cyclic resolutions on the right through the explicit equivariant diagonal of [F5] and [F8]. In characteristic two there is no Koszul sign. The two quotient-diagonal formulas contain exactly one term $e_{a} \otimes e_{b}$ for every $a + b = n$ . After pulling back the space diagonals, uniqueness of the coordinates from [F2] therefore gives

$D_{n} (x \times y) = \sum_{a + b = n} D_{a} (x) \times D_{b} (y) .$

The sum is finite, and the pure-tensor equality also shows that no comparison or Künneth splitting choice is hidden in this formula.

1.3

Reduce the coefficient $D_{q}$ and the unstable range to a sphere. [F1, F2, F9] Restriction from $K$ to its $q$ -skeleton is injective in cellular degree $q$ : if the restricted cocycle is the coboundary of a degree- $(q - 1)$ cellular cochain, the same cochain gives that coboundary on all of $K$ . Thus naturality in [F1] permits replacement of $K$ by its $q$ -skeleton.

For a cellular cocycle $c$ on a $q$ -dimensional complex, collapse the $(q - 1)$ -skeleton and map each oriented $q$ -cell to $S^{q}$ by the standard map of mod-two degree $c (e) \in {0, 1}$ . The maps agree on the collapsed boundaries, so they assemble to a specified cellular map $f : K \to S^{q}$ , and $f^{*} ι_{q} = [c]$ for the fundamental cohomology class $ι_{q}$ . Consequently $D_{q} (x) = a_{q} x$ for one scalar $a_{q} \in F_{2}$ .

The same reduction proves $D_{n} (x) = 0$ for $n > q$ . For $q < n < 2 q$ , its value on $ι_{q}$ lies in $H^{2 q - n} (S^{q}; F_{2}) = 0$ . For $n = 2 q$ and $q > 0$ , restriction to a point is an isomorphism in degree zero, while naturality and additivity give $D_{2 q} (0) = 0$ . For $n > 2 q$ the target degree is negative. When $q = 0$ , every $n > q$ already has negative target degree.

2.1

Identify the top square. [F3, step 1.1] The zero resolution coordinate is detected by restriction to a chosen augmented zero-cell of $W$ . By [F3] that restriction is $x \times x$ . Pulling it back along the diagonal of $K$ gives $D_{0} (x) = x ⌣ x$ . Since $S q_{c y c}^{q} = D_{0}$ by step 1.1, this proves the top-square identity.

3.1

Compute the scalar $a_{q}$ . [F7, step 1.2, step 1.3, step 2.1] For $q = 0$ , step 2.1 gives $D_{0} (a) = a^{2} = a$ , so $a_{0} = 1$ . For $q = 1$ , use the regular circle with vertices $A, B$ , oriented edges $J_{1}, J_{2}$ having the same boundary, fundamental cycle $J_{1} - J_{2}$ , and cocycle $u (J_{1}) = 1, u (J_{2}) = 0$ . Over $F_{2}$ , prescribe the relevant component of the carried equivariant diagonal by

$Φ (e_{1} \otimes (J_{1} - J_{2})) = J_{1} \otimes J_{1} + J_{2} \otimes J_{2} .$

Its boundary is $(A + B) \otimes (J_{1} + J_{2}) + (J_{1} + J_{2}) \otimes (A + B)$ , exactly $(1 + T) Φ (e_{0} \otimes (J_{1} - J_{2}))$ for the Alexander--Whitney diagonal. Thus this is the required component of a carried comparison, and [F7] permits its use. Evaluation by $u \otimes u$ gives $(w_{1} \times D_{1} u) (e_{1} \times (J_{1} - J_{2})) = 1$ . Hence $D_{1} u = u$ and $a_{1} = 1$ .

For $q > 1$ , take fundamental classes $u \in H^{q - 1} (S^{q - 1}; F_{2})$ and $v \in H^{1} (S^{1}; F_{2})$ . Step 1.3 kills $D_{a} (u)$ for $a > q - 1$ and $D_{b} (v)$ for $b > 1$ . Hence the coefficient $n = q$ in step 1.2 has only the split $(a, b) = (q - 1, 1)$ :

$a_{q} (u \times v) = D_{q} (u \times v) = D_{q - 1} (u) \times D_{1} (v) = a_{q - 1} a_{1} (u \times v) .$

The displayed cross product is nonzero, so $a_{q} = a_{q - 1} a_{1} = 1$ by induction. Therefore $D_{q} (x) = x$ , which is $S q_{c y c}^{0} x = x$ .

4.1

Convert the coefficient formula to Cartan. [step 1.1, step 1.2, step 1.3, step 2.1, step 3.1] Put $n = q + s - k$ in step 1.2 and set $i = q - a$ , $j = s - b$ . The vanishing from step 1.3 removes precisely the terms with $i < 0$ or $j < 0$ , and the definition in step 1.1 removes those above the input degrees. The equation $a + b = q + s - k$ is equivalent to $i + j = k$ , so

$S q_{c y c}^{k} (x \times y) = \sum_{i + j = k} S q_{c y c}^{i} (x) \times S q_{c y c}^{j} (y) .$

Pulling this equality back along the diagonal $K \to K \times K$ gives the internal formula. Step 3.1 supplies the $k = 0$ endpoint, and step 2.1 supplies the two top endpoints.

5.1

Compute the cyclic-basis action on compatible finite regular projective models. [F1, F4, F10, F11, F12, F13, F14, F15, F16, step 1.3, step 2.1, step 3.1, step 4.1] First construct the regular models needed to apply step 4.1. For $N \geq 0$ , let $L_{N}$ be the boundary complex of the $(N + 1)$ -dimensional cross-polytope. Its vertices are ${\pm e_{0}, \dots, \pm e_{N}}$ , and its faces are exactly the subsets containing no antipodal pair. Radial projection realizes $L_{N}$ as $S^{N}$ , equivariantly for the antipodal actions. Put

$Q_{N} = (sd L_{N}) / (F \sim - F) .$

This orbit object is an abstract simplicial complex, not merely a cell quotient. Indeed, a simplex of $sd L_{N}$ is a strict chain of nonempty faces by [F10]. If two such chains have the same vertex-orbits, translate one chain so that their maximal faces agree. For a face $F$ contained in that common maximal face, at most one of $F$ and $- F$ is contained there, since a face of $L_{N}$ contains no antipodal vertex pair. Thus every lower face representative, and hence the whole chain, agrees. In particular no simplex has two vertices identified, and two orbit simplices with the same vertices are equal. The quotient therefore has the closed-simplex face structure of a finite simplicial complex, so it is a finite regular cell complex.

By [F11], $∣ sd L_{N} ∣ ≅ ∣ L_{N} ∣ ≅ S^{N}$ , and the barycentric homeomorphism commutes with the antipodal action because it sends the vertex $F$ to the barycenter of $F$ . Hence

$∣ Q_{N} ∣ ≅ S^{N} / (x \sim - x) = {R P}^{N} .$

The coordinate inclusions $L_{N} \subset L_{N + 1}$ commute with antipodes; [F10] makes their subdivisions simplicial and gives compatible inclusions $Q_{N} \subset Q_{N + 1}$ . Lexicographically ordering the signed coordinate vertices and then the face chains specifies orientations of all simplices, so this construction makes no choice from a family.

Basis and product identification. The standard one-cell CW filtration of ${R P}^{\infty}$ is the quotient of the standard antipodal sphere filtration. With one lifted cell chosen in each degree, its cellular complex over $F_{2} [C_{2}]$ is the cyclic resolution $W$ : the two attaching hemispheres give alternately $T - 1$ and $1 + T$ , which are the two differentials of [F4] at $p = 2$ . Thus its quotient cochain in degree $d$ is $w_{d}$ , and [F4] identifies $t^{d} = [w_{d}]$ .

By [F12], after passing to $F_{2}$ the standard one-cell projective complex has zero differential and one generator in each degree $0 \leq d \leq N$ . The standard inclusion into the infinite filtration is the identity on every cell already present. Consequently [F13] and natural field duality [F14] show that

$H^{d} (B C_{2}; F_{2}) ⟶ H^{d} ({R P}^{N}; F_{2})$

is an isomorphism for $d \leq N$ . By [F16] it carries $t^{d}$ to the $d$ th power of the restricted class.

It remains to verify that this singular product is the cellular product used by the operation on $Q_{N}$ . Choose a cellular-to-singular chain map $j : C_{*}^{c e l l} (Q_{N}; F_{2}) \to C_{*} (∣ Q_{N} ∣; F_{2})$ carried by closed simplices. The relative carrier theorem in [F1] constructs it, and [F13] identifies its homology map with the cellular--singular comparison; [F14] therefore makes $j^{*}$ a cohomology isomorphism. The two maps

$(j \otimes j) Φ_{C} (e_{0}, -), AW Δ_{#} j$

from cellular chains of $Q_{N}$ to twofold singular chains are both augmentation-preserving and carried by the product of each closed simplex with itself. Such a carrier is augmented acyclic: each closed simplex is a disk, its singular complex contracts to a vertex, and [F15] compares the product complex with the tensor product. The carrier uniqueness clause of [F1] therefore homotopes these two maps. Dual evaluation and [F15] show that $j^{*}$ carries singular cup product to the cellular cup product appearing in step 2.1. Transporting through the homeomorphism constructed above, write $t_{N} = j^{*} (t ∣_{{R P}^{N}})$ . We have proved, for every $d \leq N$ ,

$t_{N}^{d} = j^{*} (t^{d} ∣_{{R P}^{N}}) = j^{*} ([w_{d}] ∣_{{R P}^{N}}) .$

The nonregular one-cell projective CW structure was used only for this cohomology calculation; it was not supplied to $S q_{c y c}$ .

Basis-action calculation. Fix $r, j \geq 0$ and choose $N > r + j$ . On the finite regular complex $Q_{N}$ , steps 1.3, 3.1, and 2.1 give

$S q_{c y c} (t_{N}) = S q_{c y c}^{0} (t_{N}) + S q_{c y c}^{1} (t_{N}) = t_{N} + t_{N}^{2} .$

The internal Cartan formula in step 4.1 and the basis identification above then give

$S q_{c y c} (t_{N}^{r}) = (t_{N} + t_{N}^{2})^{r} = \sum_{a = 0}^{r} (\binom{r}{a}) t_{N}^{r + a} .$

Comparing homogeneous degree $r + j$ proves $S q_{c y c}^{j} (t_{N}^{r}) = (\binom{r}{j}) t_{N}^{r + j}$ . This includes $r = 0$ , $j = 0$ , and $j > r$ . The compatible finite regular inclusions constructed above and naturality from step 1.1 make the answer independent of every larger $N$ ; the basis identification therefore permits the stable notation $S q_{c y c}^{j} (t^{r}) = (\binom{r}{j}) t^{r + j}$ .

6.1

The empty complex has zero cellular chains and all its cyclic-square coefficients vanish. [F1, F6, F14, step 1.1, step 1.3, step 2.1, step 3.1, step 4.1, step 5.1] For an empty complex, a zero cellular complex, or the zero class, all coordinates vanish. On a point only degree zero occurs, and $S q_{c y c}^{0} (a) = a^{2} = a$ . Negative and above-degree square indices are zero by definition; the degree-zero, zero-square, and top-square endpoints were calculated above. The formulas include zero factors and the unit $t^{0} = t_{N}^{0} = 1$ . The model $Q_{0}$ is a point, while every basis computation chooses $N > r + j$ , so no requested output lies above its finite model.

The cyclic operation itself uses regular cellular chains and takes no normalization quotient. The ordinary singular comparison in step 5.1 uses the unnormalized complex of [F15], so degenerate singular simplices remain present and cause no exceptional case. AC from [F6] is used in the representative and filling choices in [F1]'s equivariant-carrier comparisons and through the natural field duality [F14]. The sphere maps use the prescribed degree-zero or degree-one map on each of finitely many cells; the cross-polytope models, their orientations, every resolution diagonal and every binomial sum are explicit and finite. The singular comparison concerns only the ordinary cup product; no singular cup- $i$ comparison is asserted. ∎

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Sources

N. E. Steenrod and D. B. A. Epstein, Cohomology Operations (standard reference, not scraped)