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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

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

1.1F1F2F7F8F17A1

Record the inherited elementary properties. 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.

2.1F1F2step 1.1

Prove the top-power axiom. 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 $F_{p}$ .

2.2F1F4step 1.1

Normalize the external Cartan formula. 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.

3.1F6F8F9step 1.1step 2.1step 2.2

Calculate the operations on the cyclic coefficient algebra. 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 $0 \leq j \leq r$ .

3.2F7F11F12F13F14F15F16step 1.1step 2.2

Verify stability. 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.

4.1F1F3F5F6step 3.1

Expand the normalized double power. 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.

5.1F5step 3.1step 4.1

Prove the first relation in cofinally many degrees. 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.

5.2F5step 3.1step 4.1

Prove the second relation in cofinally many degrees. 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.

6.1F7F9F11step 2.2step 5.1step 5.2

Descend to every degree on finite regular complexes. 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.

7.1F10A1step 6.1

Pass the Adem relations to arbitrary spaces. 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.

8.1F1F2F7F8F10F11F12A1step 1.1step 2.1step 2.2step 3.1step 3.2step 4.1step 5.1step 5.2step 6.1step 7.1

Every reduced power vanishes on the empty space, and the out-of-range index conventions cover the endpoints. 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)