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Fixed point theory and framed cobordism. (English) Zbl 1047.54027

A neighbourhood retract over \(X\), or \(ENR_X\), is a space over \(X\), \(p: E \to X\), such that there is an embedding \(i: E \hookrightarrow \mathbb{R}^q \times X\) such that \(\text{proj}_X \circ i=p\), an open neighbourhood \(U\) of \(i(E)\) in \(\mathbb{R}^q \times X\), and a retraction \(r : U \to X\) such that \(p \circ r = \text{proj}_{X| U}\). Let \(p: E \to X\) be an \(ENR_X\) and let \(m, n\) be nonnegative integers. An \(m, n\)-commutative situation over \(X\) is a fiber-preserving map \(f: \mathbb{R}^n \times E \supset V \rightarrow \mathbb{R}^n \times E\) which is properly fixed. The Thom-Pontryagin construction for fixed point situations is studied and a natural correspondence between framed cobordism classes and fixed point situations is given. The fixed point situations lead to a cohomology theory, called \(FIX^*\); it generalizes to an equivariant theory for compact Lie groups. Applications to equivariant cobordism are discussed.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55M20 Fixed points and coincidences in algebraic topology
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