Vol. 103, No. 2, 1982

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On the least number of fixed points for infinite complexes

Gen Hua Shi

Vol. 103 (1982), No. 2, 377–387

Let K be a connected infinite and locally finite simplicial complex. The main theorem of this paper is the following: let L be a two-dimensionally connected infinite subcomplex of K, whose boundary L in K consists of vertices only, and f : |K|→|K| be a map. Then there exists a map F : |K|→|K|, that has the following properties: (1) Ff rel|K L|; and, (2) F has no fixed point on |L|−|L|.

The main theorem implies that if an infinite and locally finite complex K is two dimensionally connected, then the least number of fixed points of any mapping class from |K| to itself is null. At the same time, the main theorem also enables us to compute the least number m(K) of the fixed points of the identity mapping class of |K| by means of the following result: m(K) is equal to the least number n(K) of the fixed points of the good displacements of the welding set (K) of K, where (K) is the set of the boundary vertices of all these maximal two-dimensionally connected and finite subcomplexes of K.

Mathematical Subject Classification 2000
Primary: 55M20
Received: 7 August 1980
Revised: 4 February 1981
Published: 1 December 1982
Gen Hua Shi