Topological Methods in Nonlinear Analysis

Fixed point theory and framed cobordism

Carlos Prieto

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The Thom-Pontryagin construction is studied from the point of view of fixed point situations, and a very natural correspondence between framed cobordism classes and fixed point situations is given. Since fixed point classes integrate a cohomology theory, called ${\rm FIX}$, which generalizes in a natural way to an equivariant theory, this sheds light into possible approaches to equivariant cobordism.

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Topol. Methods Nonlinear Anal., Volume 21, Number 1 (2003), 155-169.

First available in Project Euclid: 30 September 2016

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Prieto, Carlos. Fixed point theory and framed cobordism. Topol. Methods Nonlinear Anal. 21 (2003), no. 1, 155--169.

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  • A. Dold, Geometric cobordism and the fixed point transfer , Algebraic Topology – Proceedings, Vancouver 1977, Lect. Notes Math., 673 , 32–87, Springer–Verlag, Berlin–Heidelberg–New York (1978) \ref\key 2 ––––, The fixed point index of fibre-preserving maps , Invent. Math., 25 (1974), 281–297 \ref\key 3 ––––, Teoría de Punto Fijo, \romVolumen I, Monografí as del Instituto de Matemáticas, 18 , UNAM, México (1984) \ref\key 4
  • T. Koźniewski, The category of submersions , Bull. Polish Acad. Sci. Math., 27 (1979), 321–326 \ref\key 5
  • C. Prieto, Coincidence index for fiber-preserving maps. An approach to stable cohomotopy , Manuscripta Math., 47 (1984), 233–249 \ref\key 6 ––––, $KO(B)$-graded stable cohomotopy over $B$ and $RO(G)$-graded $G$-equivariant cohomotopy: A fixed point theoretical approach to the Segal conjecture , Contemp. Math., 58 (1987), 89–107 \ref\key 7
  • C. Prieto and H. Ulrich, Equivariant fixed point index and fixed point transfer in nonzero dimensions , Trans. Amer. Math. Soc., 328 (1991), 731–745 \ref\key 8
  • D. Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations , Adv. Math., 7 (1971), 29–56