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) F≅frel|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 Ṁ(K) of K, where Ṁ(K) is the set of the boundary vertices of
all these maximal two-dimensionally connected and finite subcomplexes of
K.
