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Reduced powers satisfy naturality, instability, Cartan, and Adem relations

Statement

Assume AC and let $p$ be an odd prime. The normalized operations $P^{i}$ and $β P^{i}$ are natural stable additive mod- $p$ cohomology operations, of degrees $2 i (p - 1)$ and $2 i (p - 1) + 1$ , respectively. For $x \in H^{q} (X; F_{p})$ ,

P^{0} x = x, P^{i} x = 0 if 2 i > q, P^{q / 2} x = x^{p} if q is even .

They satisfy the Cartan formula

P^{k} (x ⌣ y) = \sum_{i + j = k} P^{i} (x) ⌣ P^{j} (y) .

For nonnegative integers $a, b$ with $a < p b$ , the first odd-primary Adem relation is

P^{a} P^{b} = \sum_{t = 0}^{⌊ a / p ⌋} (- 1)^{a + t} (\binom{(p - 1) (b - t) - 1}{a - p t}) P^{a + b - t} P^{t} .

For nonnegative integers $a, b$ with $a \leq p b$ , the second is

P^{a} β P^{b} = \sum_{t = 0}^{⌊ a / p ⌋} (- 1)^{a + t} (\binom{(p - 1) (b - t)}{a - p t}) β P^{a + b - t} P^{t} + \sum_{t = 0}^{⌊ (a - 1) / p ⌋} (- 1)^{a + t - 1} (\binom{(p - 1) (b - t) - 1}{a - p t - 1}) P^{a + b - t} β P^{t} .

Every binomial coefficient is reduced modulo $p$ and is zero when its lower index is negative or exceeds its nonnegative upper index. A sum with upper bound below zero is empty. Operations with negative upper index are zero.

Facts & Assumptions

Given: AC, an odd prime $p$ , the normalization $m = (p - 1) / 2$ , mod- $p$ classes, and nonnegative Adem indices $a, b$ .

[F1]

The operations $P^{i}$ and $β P^{i}$ are the normalized cyclic coefficients, with negative indices zero (Mod-p reduced power operations).

[F2]

The cyclic coefficients are natural and additive, vanish for $j < 0$ or $j > (p - 1) q$ , and satisfy $D_{0} (x) = x^{p}$ (Cyclic p-fold power construction).

[F3]

The positive-Bockstein recurrence is $β D_{2 r} = - D_{2 r - 1}$ and $β D_{2 r - 1} = 0$ (Cyclic p-fold power construction).

[F4]

The cyclic coefficients satisfy the stated odd-primary external product formula (Cyclic p-fold power construction).

[F5]

On finite regular complexes the two iterated cyclic powers have coefficients satisfying $D_{j, k} = (- 1)^{j k + p (p - 1) q / 2} D_{k, j}$ (Wreath double-power comparison and coefficient transposition).

[F6]

For odd $p$ , the cyclic coefficient algebra is $H^{*} (C_{p}; F_{p}) = F_{p} [u] \otimes Λ (v)$ , with $∣ u ∣ = 2$ , $∣ v ∣ = 1$ , and $u = β v$ (Free cyclic resolution, group cohomology, and cochain transfer).

[F7]

The Bockstein is natural and commutes with the signed reduced cohomology suspension (Bocksteins are natural and stable).

[F8]

The Bockstein is computed by lifting a cocycle and dividing its coboundary through the coefficient injection (Bockstein connecting operation).

[F9]

The mod- $p$ Bockstein satisfies the signed product derivation rule (The mod-two Bockstein is a derivation).

[F10]

A natural mod- $p$ identity valid on every finite regular complex is valid on every space (Natural singular-cohomology identities are detected on finite regular complexes).

[F11]

Under AC, cross product with the circle generator is injective by the cohomological Kunneth isomorphism (Cohomological Kunneth cross product is a ring isomorphism).

[F12]

Stability means commutation with the signed reduced cohomology suspension (Stable natural cohomology operation).

[F13]

For a well-pointed based space $(X, x_{0})$ , form the reduced cone $C X = (X \times I) / (X \times {1} \cup {x_{0}} \times I)$ and its quotient by the height-zero base, $Σ X = C X / X$ .

[F14]

The cone-pair connecting map sends a cocycle to the coboundary of an extension (Long exact sequence of a pair in singular cohomology).

[F15]

Homotopic maps induce equal cohomology maps for every abelian coefficient group (Homotopic maps induce equal maps in singular cohomology).

[F16]

Excision identifies relative cohomology after removing a closed subset lying in the interior of the relative subspace (Excision for singular cohomology).

[F17]

The top cyclic coefficient is $D_{(p - 1) q} (x) = (- 1)^{m q (q + 1) / 2} (m!)^{q} x$ (Cyclic p-fold power construction).

[A1]

The Axiom of Choice supplies exactly the choices already exposed by [F1]--[F5], [F10], and the additive Kunneth isomorphism [F11].

Proof

Proof technique: normalize the cyclic coefficients, calculate their Cartan and top values, expand the two-stage cyclic power coefficient by coefficient, apply the row--column symmetry and Lucas reduction, then descend from cofinally many degrees with the circle generator.

1.1

Record the inherited elementary properties. [F1, F2, F7, F8, F17, A1] The degree formulas and negative-index convention follow from [F1]. If $2 i > q$ , the cyclic index in [F1] is negative, so [F2] proves strict instability. At $i = 0$ , substitution of [F17] in [F1] gives

$P^{0} (x) = (- 1)^{m q (q + 1)} (m!)^{- q} (m!)^{q} x = x .$

Naturality and additivity of $P^{i}$ follow from [F2]. For two cocycles, the sum of chosen lifts is a lift of their sum and its coboundary is the sum of their coboundaries, so [F8] makes the Bockstein additive; its naturality is [F7]. Thus every $β P^{i}$ is natural and additive as well.

2.1

Prove the top-power axiom. [F1, F2, step 1.1] Finite inverse-pairing in $F_{p}^{\times}$ gives Wilson's identity: every element other than $1, - 1$ cancels with its distinct inverse, so $(p - 1)! = - 1$ . Pairing $k$ with $p - k$ , $1 \leq k \leq m$ , also gives

(p - 1)! = (- 1)^{m} (m!)^{2}, (m!)^{2} = (- 1)^{m + 1} .

If $q = 2 i$ , the cyclic index in [F1] is zero and [F2] gives $D_{0} (x) = x^{p}$ . The scalar multiplying it is

(- 1)^{i + m (4 i^{2} + 2 i) / 2} (m!)^{- 2 i} = (- 1)^{i (m + 1)} (- 1)^{i (m + 1)} = 1.

Hence $P^{q / 2} (x) = x^{p}$ . The case $q = 0 = i$ agrees with $P^{0} = i d$ because $a^{p} = a$ in $F_{p}$ .

2.2

Normalize the external Cartan formula. [F1, F4, step 1.1] Let $x, y$ have degrees $r, s$ . In the even cyclic coordinate $(r + s - 2 k) (p - 1)$ , [F4] leaves precisely the splits $(r - 2 i) (p - 1) + (s - 2 j) (p - 1)$ with $i + j = k$ . Substitute the definition [F1] into [F4]'s external formula. Factorials cancel. The total sign exponent modulo two is

k + i + j + m (r^{2} + r + s^{2} + s + r s) + p m r s .

Here $i + j = k$ , both $r^{2} + r$ and $s^{2} + s$ are even, and $p + 1$ is even, so this exponent is even. Therefore

P^{k} (x \times y) = \sum_{i + j = k} P^{i} (x) \times P^{j} (y) .

Pullback along the diagonal gives the asserted internal Cartan formula. Every sum is finite by instability.

3.1

Calculate the operations on the cyclic coefficient algebra. [F6, F8, F9, step 1.1, step 2.1, step 2.2] By degree, instability, and the top-power axiom, $P^{0} v = v$ , $P^{i} v = 0$ for $i > 0$ , $P^{0} u = u$ , $P^{1} u = u^{p}$ , and $P^{i} u = 0$ for $i > 1$ . Repeated Cartan expansion thus gives, for $r, j \geq 0$ ,

P^{j} (u^{r}) = (\binom{r}{j}) u^{r + j (p - 1)}, P^{j} (v u^{r}) = (\binom{r}{j}) v u^{r + j (p - 1)} .

Also $β u = 0$ : if an integral lift of a cocycle for $v$ has coboundary $p h$ , then $h$ is itself a cocycle and is an integral lift of $β v$ , so its Bockstein is zero by [F8]. The derivation rule [F9] now gives

β P^{j} (u^{r}) = 0, β P^{j} (v u^{r}) = (\binom{r}{j}) u^{r + 1 + j (p - 1)} .

These are exactly the four even/odd coefficient actions used in the double power calculation, with a binomial declared zero outside $0 \leq j \leq r$ .

3.2

Verify stability. [F7, F11, F12, F13, F14, F15, F16, step 1.1, step 2.2] Let $z$ generate ${\tilde{H}}^{1} (S^{1}; F_{p})$ . The cone-pair quotient comparison needs proof. For a based CW $X$ , radially subdivide the one open cell containing the basepoint if needed, retaining the higher attaching maps; this finite refinement makes it a vertex without changing the based space. Each nonbasepoint $n$ -cell produces an $(n + 1)$ -cell from its product with the open cone-height interval, with the height-zero cells forming $X$ and the height-one face and basepoint track collapsed to one vertex. Product characteristic disks have finite boundary-cell support, and their quotient map-out and weak-topology tests assemble a CW structure on $C X$ with $X$ a closed subcomplex. Its cellwise radial collar is an open neighborhood $V$ strongly deformation retracting onto $X$ , with the characteristic-disk flows assembled by the CW weak topology. Since $V$ contains the entire fibre $X$ collapsed by $q_{C} : C X \to Σ X$ , it is saturated, so $q_{C} (V)$ is open and retracts to the quotient vertex. The pair sequences and [F15] make $H^{*} (V, X; F_{p})$ and $H^{*} (q_{C} (V), {*}; F_{p})$ vanish. The short exact cochain sequences for the corresponding triples, surjective by zero extension, replace $X$ by $V$ and the vertex by $q_{C} (V)$ in relative cohomology. By [F16], excise $X$ and the quotient vertex. The remaining pairs are homeomorphic under the quotient map, so $q_{C}^{*} : H^{*} (Σ X, {*}; F_{p}) \to H^{*} (C X, X; F_{p})$ is an isomorphism. The connector [F14] followed by its inverse is the standard cohomology suspension.

Represent $x$ by a relative cocycle $a$ on $(X, {x_{0}})$ , and let $v$ be the interval endpoint $0$ -cochain whose coboundary represents the oriented interval class. Extending $a$ across the cone by the interval cutoff $v$ , the positive coboundary and the signed external product rule give $(- 1)^{∣ x ∣} a \times δ v$ as the cone-pair connector representative [F11, F14]. Identifying the two-ended interval quotient with $S^{1}$ and $Σ X$ with $X \land S^{1}$ , the stable convention $σ_{n} = (- 1)^{n} (q_{C}^{*})^{- 1} \partial$ from [F7] cancels this coboundary sign. Thus the signed suspension obeys $q_{S}^{*} (σ x) = x \times z$ for $q_{S} : X \times S^{1} \to X \land S^{1} = Σ X$ . The same cone-pair calculation and [F11] show that $q_{S}^{*}$ is injective on reduced cohomology: under Kunneth, its suspension summand is exactly cross product with $z$ .

Instability gives $P^{0} z = z$ and $P^{j} z = 0$ for $j > 0$ . Since $P^{i}$ is $F_{p}$ -linear, the external Cartan formula yields $q_{S}^{*} P^{i} (σ x) = q_{S}^{*} σ (P^{i} x)$ ; the suspension signs agree because $P^{i}$ has even degree. Injectivity gives $P^{i} σ = σ P^{i}$ . Thus $P^{i}$ is stable in the sense of [F12], and [F7] makes the composite $β P^{i}$ stable as well.

4.1

Expand the normalized double power. [F1, F3, F5, F6, step 3.1] Put $λ (q) = (- 1)^{m (q^{2} + q) / 2} (m!)^{- q}$ . The definition and [F3]'s positive-Bockstein identity rewrite the diagonal cyclic power of a degree $q$ class as

λ (q) d^{*} P (x) = \sum_{i} (- 1)^{i} (w_{(q - 2 i) 2 m} \times P^{i} x - w_{(q - 2 i) 2 m - 1} \times β P^{i} x) .

Apply the same normalized expansion once more, use step 3.1 on each $w$ -coordinate, and use Cartan to expand the products. This produces four finite coefficient rows: even--even, even--odd, odd--even, and odd--odd. Interchanging the two resolution factors changes a coefficient by the exact sign $(- 1)^{j k + p (p - 1) q / 2} = (- 1)^{j k + m q}$ in [F5], where the original input has degree $q$ . We use this row comparison below only when $q = Q$ is even, so the global factor $(- 1)^{m q}$ is $1$ . The even--even and mixed even--odd rows then have $j k$ even and transposition sign $1$ ; the odd--odd row has sign $- 1$ and is not used in either displayed relation. In each mixed row, the minus sign in the positive-Bockstein expansion is retained on both sides, giving the stated $β$ -Adem coefficients after normalization. Coordinate uniqueness in [F5] therefore reduces the two relations, on even-degree inputs, to the binomial comparisons in the next steps. No coefficient comparison for odd $q$ is claimed here; a later circle-descent argument extends the resulting identities to odd degrees.

5.1

Prove the first relation in cofinally many degrees. [F5, step 3.1, step 4.1] Fix $a < p b$ and choose $s$ with $p^{s} > a$ . Set

Q = 2 (1 + p + \dots + p^{s - 1}) + 2 b .

In the even--even coefficient comparison of step 4.1, the binomial $(\binom{(Q - 2 i) m}{i - b})$ is zero unless $i = b$ and is one at $i = b$ . The transposed coefficient indexed by $t$ is

(\binom{(Q - 2 t) m}{a - p t}) = (\binom{p^{s} - 1 + (p - 1) (b - t)}{a - p t}) .

Thus the row comparison is the actual identity

P^{a} P^{b} (x) = \sum_{t = 0}^{⌊ a / p ⌋} (- 1)^{a + t} (\binom{(Q - 2 t) m}{a - p t}) P^{a + b - t} P^{t} (x) .

Only $0 \leq t \leq ⌊ a / p ⌋$ occur. Since $a < p b$ , each such $t < b$ , and since $a - p t < p^{s}$ , the base- $p$ binomial expansion (or coefficient comparison in $(1 + T)^{p^{s} - 1 + N} = (1 + T^{p^{s}}) (1 + T)^{N - 1}$ ) gives

(\binom{p^{s} - 1 + (p - 1) (b - t)}{a - p t}) = (\binom{(p - 1) (b - t) - 1}{a - p t}) in F_{p} .

The normalization and row--column sign in step 4.1 contribute $(- 1)^{a + t}$ . Hence every degree- $Q$ class on a finite regular complex satisfies the first displayed Adem relation.

5.2

Prove the second relation in cofinally many degrees. [F5, step 3.1, step 4.1] Fix $a \leq p b$ , choose $s$ with $p^{s} > a$ , and now set $Q = 2 p^{s} + 2 b$ . The even--odd and odd--even coefficient rows of step 4.1 select the unique left-hand term $P^{a} β P^{b}$ . On the transposed side their two coefficients are

(\binom{(Q - 2 t) m}{a - p t}) = (\binom{(p - 1) (p^{s} + b - t)}{a - p t})

and

(\binom{(Q - 2 t) m - 1}{a - p t - 1}) = (\binom{(p - 1) (p^{s} + b - t) - 1}{a - p t - 1}) .

Consequently the two mixed rows give, before reduction,

P^{a} β P^{b} (x) = \sum_{t = 0}^{⌊ a / p ⌋} (- 1)^{a + t} (\binom{(Q - 2 t) m}{a - p t}) β P^{a + b - t} P^{t} (x) + \sum_{t = 0}^{⌊ (a - 1) / p ⌋} (- 1)^{a + t - 1} (\binom{(Q - 2 t) m - 1}{a - p t - 1}) P^{a + b - t} β P^{t} (x) .

Because both lower indices are below $p^{s}$ , the same base- $p$ coefficient comparison reduces these to $(\binom{(p - 1) (b - t)}{a - p t})$ and $(\binom{(p - 1) (b - t) - 1}{a - p t - 1})$ , respectively. The first lower index is nonnegative exactly through $t = ⌊ a / p ⌋$ ; the second exactly through $t = ⌊ (a - 1) / p ⌋$ . Tracking the normalized odd row gives the signs $(- 1)^{a + t}$ and $(- 1)^{a + t - 1}$ . Thus every degree- $Q$ class on a finite regular complex satisfies the second displayed relation.

6.1

Descend to every degree on finite regular complexes. [F7, F9, F11, step 2.2, step 5.1, step 5.2] Let $R$ be the residual of either relation and suppose it vanishes on degree- $r$ classes. For a degree- $(r - 1)$ class $x$ , form $x \times z$ , with $z$ the circle generator. Cartan and instability give $P^{j} (x \times z) = P^{j} x \times z$ . Moreover $β z = 0$ , since $H^{2} (S^{1}; F_{p}) = 0$ , so the derivation rule [F9] gives $β P^{j} (x \times z) = β P^{j} x \times z$ . Applying Cartan once again to every composite in $R$ yields

R (x \times z) = R (x) \times z .

The left side is zero, while [F11] makes cross product with $z$ injective. Thus $R (x) = 0$ . The degrees $Q$ in steps 5.1 and 5.2 are unbounded as $s$ grows, so finite iteration descends to every nonnegative input degree.

7.1

Pass the Adem relations to arbitrary spaces. [F10, A1, step 6.1] For fixed $p, a, b$ , either residual is a natural additive map between fixed singular cohomology degrees by step 1.1. Step 6.1 makes it zero on every finite regular complex. The detector [F10] therefore makes it zero on every space. This proves both Adem relations globally.

8.1

Every reduced power vanishes on the empty space, and the out-of-range index conventions cover the endpoints. [F1, F2, F7, F8, F10, F11, F12, A1, step 1.1, step 2.1, step 2.2, step 3.1, step 3.2, step 4.1, step 5.1, step 5.2, step 6.1, step 7.1] The empty space and zero class give zero throughout. On a point, step 1.1 leaves $P^{0} = i d$ in degree zero and all positive operations zero; step 2.1 includes both zero and one. The top endpoint $2 i = q$ , the strict range $2 i > q$ , and negative operations are explicit.

For $a = 0$ , the first relation (when $b > 0$ ) is $P^{0} P^{b} = P^{b} P^{0}$ . In the second relation the first sum has only $t = 0$ , while the second is empty; at $a = b = 0$ this reads $β = β$ . At $a = p b$ , included only in the second relation, the stated zero-binomial convention controls both terminal terms. Each finite sum includes both endpoints, and the hypotheses $a < p b$ and $a \leq p b$ were used exactly in steps 5.1 and 5.2.

Degenerate singular simplices are already included by [F1] and [F2]. Step 3.2 treats reduced degree zero and the one-point based space. No biconditional is asserted. AC is assumed and propagated exactly through the cyclic and wreath carriers, the finite detector, and the additive Kunneth isomorphism; the Wilson pairing, binomial coefficient extractions, circle descent, and all sums are finite. ∎

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Sources

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