The Lefschetz fixed point theorem for noncompact locally connected spaces

Knill, R. J.

doi:10.1090/S0002-9947-1972-0315700-9

The Lefschetz fixed point theorem for noncompact locally connected spaces
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Trans. Amer. Math. Soc. 174 (1972), 185-198 Request permission

Abstract:

Leray’s notion of convexoid space is localized and used to show that if $f:M \to M$ is a relatively compact map on a locally convex manifold M, and f has no fixed points then its Lefschetz trace is zero. A similar theorem holds for certain adjunction spaces $Y{ \cup _g}Z$ where Y is Q-simplicial and Z is locally convexoid. A number of other properties of locally convexoid spaces are derived; for example, any neighborhood retract of a locally convexoid space is locally convexoid.

References

Topologie

Edward G. Begle, A fixed point theorem, Ann. of Math. (2) 51 (1950), 544–550. MR 35433, DOI 10.2307/1969367
R. H. Bing, Challenging conjectures, Amer. Math. Monthly 74 (1967), no. 1, 56–64. MR 203661, DOI 10.2307/2314868
Armand Borel, The Poincaré duality in generalized manifolds, Michigan Math. J. 4 (1957), 227–239. MR 98382

Sur un continu acyclique qui se laisse transformer topologiquement en lui meme sans points invarients

24

Thomas R. Brahana, Products of quasi-complexes, Proc. Amer. Math. Soc. 7 (1956), 954–948. MR 80920, DOI 10.1090/S0002-9939-1956-0080920-4
Felix E. Browder, On the fixed point index for continuous mappings of locally connected spaces, Summa Brasil. Math. 4 (1960), 253–293 (1960). MR 146834
Felix E. Browder, Fixed point theorems on infinite dimensional manifolds, Trans. Amer. Math. Soc. 119 (1965), 179–194. MR 195082, DOI 10.1090/S0002-9947-1965-0195082-2

Sur la connexité local d’ordre supérieur

2

C. H. Dowker, Homology groups of relations, Ann. of Math. (2) 56 (1952), 84–95. MR 48030, DOI 10.2307/1969768
James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
James Eells Jr., A setting for global analysis, Bull. Amer. Math. Soc. 72 (1966), 751–807. MR 203742, DOI 10.1090/S0002-9904-1966-11558-6
Olof Hanner, Retraction and extension of mappings of metric and non-metric spaces, Ark. Mat. 2 (1952), 315–360. MR 50875, DOI 10.1007/BF02591501
Shin’ichi Kinoshita, On some contractible continua without fixed point property, Fund. Math. 40 (1953), 96–98. MR 60225, DOI 10.4064/fm-40-1-96-98
Victor Klee, Leray-Schauder theory without local convexity, Math. Ann. 141 (1960), 286–296. MR 131150, DOI 10.1007/BF01360763
R. J. Knill, $Q$-simplicial spaces, Illinois J. Math. 14 (1970), 40–51. MR 258017
Ronald J. Knill, Cones, products and fixed points, Fund. Math. 60 (1967), 35–46. MR 211389, DOI 10.4064/fm-60-1-35-46

Algebraic topology

4

Jean Leray, Sur la forme des espaces topologiques et sur les points fixes des représentations, J. Math. Pures Appl. (9) 24 (1945), 95–167 (French). MR 15786
Jean Leray, Théorie des points fixes: indice total et nombre de Lefschetz, Bull. Soc. Math. France 87 (1959), 221–233 (French). MR 143202
Jean Leray, La théorie des points fixes et ses applications en analyse, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, Amer. Math. Soc., Providence, R.I., 1952, pp. 202–208 (French). MR 0047318
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
Richard B. Thompson, On the semicomplexes of F. Browder, Bull. Amer. Math. Soc. 73 (1967), 531–536. MR 214050, DOI 10.1090/S0002-9904-1967-11733-6
Richard B. Thompson, A unified approach to local and global fixed point indices, Advances in Math. 3 (1969), 1–71. MR 247627, DOI 10.1016/0001-8708(69)90002-4
John W. Tukey, Convergence and Uniformity in Topology, Annals of Mathematics Studies, No. 2, Princeton University Press, Princeton, N. J., 1940. MR 0002515
Raymond Louis Wilder, Topology of Manifolds, American Mathematical Society Colloquium Publications, Vol. 32, American Mathematical Society, New York, N. Y., 1949. MR 0029491