ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-14

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Thom spaces of trivial line and plane bundles

Example

For the standard oriented trivial real bundles, $Th (ε_{B}^{1}) = Σ B_{+}; Th (ε_{B}^{2}) = Σ^{2} B_{+} .$ Their Thom classes are the once- and twice-suspended units, and their Thom isomorphisms are the corresponding relative suspension isomorphisms.

Facts & Assumptions

Given: A space $B$ , a commutative ring $R$ , and the standard ordered orientations of $R$ and $R^{2}$ .

[F1]

Thom spaces of zero and trivial bundles calculates the Thom space of a trivial rank- $n$ bundle as $Σ^{n} B_{+}$ .

[F2]

Thom isomorphism for a trivial oriented bundle constructs its normalized class by the ordered relative suspension and proves cup by it is an isomorphism over arbitrary $R$ without AC.

Verification

technique · substitute ranks one and two

1.1

Put $n = 1$ in [F1]. The pair is $(B \times [- 1, 1], B \times {- 1, 1})$ , its quotient is $Σ B_{+}$ , and [F2]'s fiber generator is the connector of $(0, 1)$ on the two boundary endpoints. Thus its pullback to the product is the suspended unit and cup by it is the one-fold relative suspension isomorphism.

F1F2

1.2

Put $n = 2$ . The quotient is $Σ^{2} B_{+}$ , the ordered generator is the second connector applied to the rank-one generator, and [F2] gives the twice-iterated suspension isomorphism. This fixes the orientation sign rather than choosing an unspecified generator.

F1F2

2.1

For empty $B$ the spaces are the one-point based space and cohomology maps are zero; for a point base the spaces are $S^{1}$ and $S^{2}$ . The zero ring, zero/unit classes, both interval endpoints, the two coordinate orders and both suspension endpoints are explicit in steps 1.1–1.2. All connectors are finite and formulaic, so no AC is used.

F1F2step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

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