## 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

### Keywords:

Cobordism; fixed point index; generalized cohomology
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