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Stable Steenrod squares on universal Thom cohomology

Statement

Assume AC, inherited from the cited bundle, cohomology, or operation suppliers. For $i \geq 0$ and $x = (x_{r}) \in {\hat{H}}^{q} (T O; F_{2})$ , define $S q^{i} x$ by the components $(S q^{i} x)_{r} = S q^{i} (x_{r})$ in ${\tilde{H}}^{r + q + i} (T_{r}; F_{2})$ . At a negative finite-level source degree use the unique map from the zero cohomology group. These components form a compatible tuple, giving a linear map $S q^{i} : {\hat{H}}^{q} \to {\hat{H}}^{q + i}$ . They satisfy $S q^{0} = i d$ and the published Adem relations, so their finite linear combinations and composites give an action of the mod-two square algebra on the graded invariant. For the stable normalized Thom vector $U$ , $S q^{i} U = w_{i} U$ with $w_{0} = 1$ ; its rank- $r$ component is zero for $i > r$ . The degree-zero stable vector $U$ is not subject to the instability bound for a degree-zero class of a space: its rank- $r$ representative has degree $r$ . No freeness or homotopy-detection conclusion is asserted here.

Facts & Assumptions

Given: AC; the degreewise inverse-limit module ${\hat{H}}^{*} (T O; F_{2}) = F_{2} [w_{1}, w_{2}, \dots] \cdot U$ of Degreewise mod-two cohomology of the universal real Thom prespectrum with its component classes $u_{n}$ ; a stable class $x = (x_{n})$ ; and the componentwise square operations $S q^{i} (x)_{n} = S q^{i} (x_{n})$ .

[F1]

The degreewise constancy lemma identifies the inverse limit with the polynomial Thom module and its componentwise identification, and the structure-map naturality used below is the one recorded there (Stable universal Thom cohomology is eventually constant in every degree).

[F2]

Squares are natural additive operations on cohomology, commute with the cohomology suspension, and satisfy the Thom identity $S q^{i} u_{n} = w_{i} (γ_{n}) u_{n}$ (Steenrod squares are well-defined and natural, Steenrod normalization, instability, suspension, and top square, Thom identity for Stiefel–Whitney classes); the Adem relations and admissible calculus act on the limit module (Adem relations for Steenrod squares, The mod-two square algebra, admissible sequences, and excess).

Proof

technique · direct

1.1givenF1F2

Naturality commutes squares with αₙ*, and the published square-suspension theorem commutes them with σ and its inverse. Thus ρₙ(q+i)Sqⁱ=Sqⁱρₙ(q), proving compatibility. The Thom identity at rank n gives Sqⁱuₙ=wᵢ(γₙ)uₙ, with both sides zero for i>n. These are precisely the components of wᵢU under the preceding polynomial description.

2.1step 1.1F2∎

For n=0, Sq⁰u₀=u₀ and higher squares vanish. No unstable top-square or degree-zero instability is asserted for the stable class U: its component uₙ has degree n, and those unstable bounds depend on n. Iterated words of squares and their already-proved Adem relations therefore act on this limit module. This does not prove its freeness as a Steenrod module, compute the Steenrod algebra's basis, or prove Hurewicz injectivity. Those remain distinct supplier obligations.

Depends on

Used by

Dependency tree · two levels

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Sources

Tom Weston, An Introduction to Cobordism Theory (standard reference, not scraped)
John Milnor and James Stasheff, Characteristic Classes (standard reference, not scraped)