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Stable universal Thom cohomology is eventually constant in every degree

Statement

Assume AC through the published Thom and universal-bundle suppliers. For the preceding compatible-tuple invariant, the normalized Thom isomorphisms identify $ρ_{r} (q)$ with the homogeneous-weight- $q$ map $F_{2} [w_{1}, \dots, w_{r + 1}]^{q} \to F_{2} [w_{1}, \dots, w_{r}]^{q}$ sending $w_{i}$ to $w_{i}$ for $i \leq r$ and $w_{r + 1}$ to zero. If $q < 0$ all terms are zero. If $q \geq 0$ , every bonding map with $r \geq q$ is an isomorphism, and projection to any level $r \geq q$ identifies ${\hat{H}}^{q} (T O; F_{2})$ with $F_{2} [w_{1}, \dots, w_{r}]^{q} u_{r}$ . Writing $U = (u_{r})$ , this gives ${\hat{H}}^{*} (T O; F_{2}) ≅ F_{2} [w_{1}, w_{2}, \dots] U$ , with $∣ w_{i} ∣ = i$ and $∣ U ∣ = 0$ , as a graded vector space and as a module over the stable characteristic polynomial algebra. Each homogeneous piece is finite-dimensional. The assertion is a Thom-module computation, not an identification of reduced cup rings or a general spectrum-comparison theorem.

Facts & Assumptions

Given: AC; the fixed-coordinate Thom prespectrum of The Thom prespectrum of the universal real and oriented bundles with structure maps $α_{n}$ and normalization $h_{n}^{*} u_{ε \oplus γ_{n}} = σ (u_{n})$ from its construction; the degreewise inverse system $A_{n} (q) = {\tilde{H}}^{n + q} (T_{n}; F_{2})$ with transition $ρ_{n} (q) = σ^{- 1} α_{n}^{*}$ ; and the compatible-tuple invariant of Degreewise mod-two cohomology of the universal real Thom prespectrum, whose polynomial presentation is to be proved.

[F1]

The canonical mod-two orientations (R-oriented vector bundle and orientation local system) license the Thom isomorphisms identifying ${\tilde{H}}^{n + q} (T_{n}; F_{2})$ with $H^{q} (B O (n); F_{2})$ via $a \mapsto a u_{n}$ ; its polynomial presentation is supplied by [F2], and the structure maps satisfy the naturality and normalization identities recorded in the prespectrum definition (The Thom prespectrum of the universal real and oriented bundles, Thom isomorphism for oriented vector bundles, The Thom quotient identifies relative and reduced cohomology).

[F2]

The mod-two cohomology of $B O (n)$ is the polynomial ring $F_{2} [w_{1}, \dots, w_{n}]$ on the universal Stiefel–Whitney classes, and pullback along stabilization fixes $w_{i}$ for $i \leq n$ and kills $w_{i}$ for $i > n$ by the Whitney formula and naturality (Mod-two cohomology of BO(n), Whitney sum formula for Stiefel–Whitney classes, Naturality of Stiefel–Whitney classes).

[F3]

AC is inherited from the Thom and universal characteristic-class suppliers in [F1]–[F2]; the module presentation is proved below by their bonding maps and unique extension of compatible tuples (The Axiom of Choice).

Proof

technique · direct

1.1givenF1F2

Write Φₙ(a)=a uₙ. Pullback naturality and the normalization in the prespectrum definition give αₙΦₙ₊₁(a)=σΦₙ(sₙa). Thus Φₙ⁻¹ρₙΦₙ₊₁=sₙ*, checking every domain and degree. The bundle isometry gives sₙγₙ₊₁=ε¹⊕γₙ; Whitney and naturality imply sₙwᵢ=wᵢ for i≤n and zero above n. The published BO(n) polynomial theorem then gives exactly the displayed map. Negative base cohomology vanishes. In weight q no variable of weight greater than q occurs, so n≥q makes the transition bijective. A compatible tuple is therefore uniquely determined by one value in the constant tail, and every such value extends uniquely forward through inverse isomorphisms and backward through the specified maps.

2.1step 1.1F2F3∎

Homogeneous weight-q polynomials in infinitely many generators are exactly that tail. There are finitely many partitions of q, since each exponent satisfies 0≤aᵢ≤q/i and only i≤q occurs, proving finiteness. U is compatible by the normalization identity.

Depends on

Used by

Cited to discharge well-definedness by Degreewise mod-two cohomology of the universal real Thom prespectrum.

Dependency tree · two levels

69 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Tom Weston, An Introduction to Cobordism Theory (standard reference, not scraped)
J. P. May, A Concise Course in Algebraic Topology (standard reference, not scraped)