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Universal real Thom spaces are (r−1)-connected

Statement

Assume AC, inherited from the cited bundle, cohomology, or operation suppliers. For each r≥2, T_r=Th(γ_r over BO(r)) is a nonempty (r−1)-connected based CW complex.

Facts & Assumptions

Given: AC; the based CW Thom space $T_{r} = T h (γ_{r})$ of the universal real rank- $r$ bundle from The Thom prespectrum of the universal real and oriented bundles, with its Schubert-cell CW structure and basepoint vertex.

[F1]

The prespectrum definition constructs $T_{r}$ as a based CW complex with the weak cell topology and basepoint vertex (The Thom prespectrum of the universal real and oriented bundles).

[F2]

The Schubert cells of $B O (r)$ attach with finite boundary support and give the CW structure; over a $d$ -cell the Thom construction contributes a cell of dimension $d + r$ (Schubert cells give the stable Grassmannian CW structure, The Thom prespectrum of the universal real and oriented bundles).

[F3]

For a CW pair whose relative cells all have dimension at least $n + 1$ , the inclusion of the subcomplex induces isomorphisms on $π_{i}$ for $i \leq n - 1$ and a surjection on $π_{n}$ (High relative cells do not change lower homotopy).

Proof

technique · direct

1.1givenF1F2

The prespectrum construction constructs T_r as a based CW complex. Over a d-dimensional Schubert cell of BO(r), its Thom attachment contributes cells of dimension d+r, with d≥0; hence every cell outside the basepoint has dimension at least r. Apply the published high-relative-cells lemma to the CW pair (T_r,*): it is (r−1)-connected. In particular T_r is path-connected and simply connected for r≥2. The basepoint is a vertex.

2.1step 1.1F3∎

This is the separate connectivity argument required by the finite-range comparison; it does not follow from a cohomological Thom isomorphism alone.

Depends on

Used by

Dependency tree · two levels

46 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

Allen Hatcher, Algebraic Topology (standard reference, not scraped)