A method of combining fixed points

Author:
Roger Waggoner

Journal:
Proc. Amer. Math. Soc. **51** (1975), 191-197

MSC:
Primary 54H25; Secondary 57C05

MathSciNet review:
0402713

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Abstract: It is now well known that in the category of finite polyhedra the fixed point property is not preserved by the operations of suspension, Cartesian product, adjunction along a segment, and join. Thus far none of the examples given have involved polyhedra of dimension 2. It is shown in this paper that two fixed points and of a self-map of a polyhedron can be combined in a certain way if a certain criterion is satisfied by the -image of a path from to . Several corollaries follow, one of which is that if is a finite simply connected -polyhedron with no local separating points, , and has a -simplex such that is cyclic, then fails to have the fixed point property. This eliminates many -dimensional polyhedra from consideration as examples.

**[1]**R. H. Bing,*The elusive fixed point property*, Amer. Math. Monthly**76**(1969), 119–132. MR**0236908****[2]**Robert F. Brown,*The Lefschetz fixed point theorem*, Scott, Foresman and Co., Glenview, Ill.-London, 1971. MR**0283793****[3]**Edward Fadell,*Recent results in the fixed point theory of continuous maps*, Bull. Amer. Math. Soc.**76**(1970), 10–29. MR**0271935**, 10.1090/S0002-9904-1970-12358-8**[4]**Shi Gen-hua,*On least number of fixed points and Nielsen numbers*, Chinese Math.–Acta**8**(1966), 234–243. MR**0210109****[5]**Roger Waggoner,*A fixed point theorem for (𝑛-2)-connected 𝑛-polyhedra*, Proc. Amer. Math. Soc.**33**(1972), 143–145. MR**0293622**, 10.1090/S0002-9939-1972-0293622-5

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DOI:
https://doi.org/10.1090/S0002-9939-1975-0402713-X

Keywords:
Fixed point property,
Lefschetz number,
fixed point index

Article copyright:
© Copyright 1975
American Mathematical Society