Geometric Properties Of Piecewise Linear Maps From A Simplicial Complex To Euclideam Spaces

Piecewise linear map and piecewise linear embedding from a simplicial complex to Euclidean space are important concepts in algebraic topplogy.In this paper.We investigate some geomtric properties of piecewise linear maps.The paper is organized in two parts.The first part concerns a conclusion(without proof)appearing in a paper by Gromov and Guth in 2012.We give a detailed proof of this conclusion.Precisely,if K is a finite k-dimensional simplicial complex,f:|K|?RN is a continuous map and n?2k+1,then for any ?>0,then there is a piecewise linear embedding g:|K|?Rn such that |g(v)-f(v)|<? for any vertex v of K.The second part concerns the Lipschitzness of a piecewise linear map.The result is as follows:If K is a k-dimensional simplicial complex and f:|K|?Rn is a piecewise linear map,then f is a Lipschitz map with respect to the barycentric coordinate metric or polyhedral metric on |K|.An estimate of Lipschitz constant of f is also given.