PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Thom spaces of zero and trivial bundles

Statement

Naturally in $B$ , $Th (0_{B}) = B_{+}; Th (B \times R^{n}) = B_{+} \land S^{n} = Σ^{n} B_{+},$ and the trivial disk/sphere pair is $(B \times D^{n}, B \times S^{n - 1})$ .

Facts & Assumptions

Given: A space $B$ , the zero bundle, and the product bundle with its standard Euclidean metric.

[F1]

Disk, sphere, and Thom spaces of a metric vector bundle defines the disk, sphere, and based quotient, including the empty-sphere convention in rank zero.

Proof

technique · calculate the defining quotients

1.1

In rank zero each fiber is the singleton zero vector, so [F1] gives $D (0_{B}) = B$ and $S (0_{B}) = \emptyset$ . By the based-quotient convention, $Th (0_{B}) = B_{+}$ .

1.2

For $B \times R^{n}$ with the product metric, the norm depends only on the second coordinate, hence $(D, S) = (B \times D^{n}, B \times S^{n - 1})$ . Collapsing the second subspace gives $(B \times D^{n}) / (B \times S^{n - 1}) ≅ B_{+} \land (D^{n} / S^{n - 1}) = B_{+} \land S^{n}$ .

2.1

By definition $B_{+} \land S^{n} = Σ^{n} B_{+}$ . Every displayed map sends $(b, v)$ by the identity formula and therefore commutes with pullback along a map $B^{'} \to B$ . For $B = \emptyset$ both sides are the one-point based space; for $n = 0$ , $S^{- 1} = \emptyset$ and step 1.2 reduces to step 1.1; for $n = 1$ the boundary consists of the two endpoints. Zero and unit radii and the quotient basepoint are preserved, and no choice is used.

F1step 1.1step 1.2∎

Depends on

Disk, sphere, and Thom spaces of a metric vector bundle

Used by

Thom spaces of trivial line and plane bundles Example
Thom isomorphism for a trivial oriented bundle Theorem

Dependency tree · two levels

2 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

May, A Concise Course in Algebraic Topology, Chapter 23 §5 (standard reference, not scraped)