Simple Homotopy Types and Finite Spaces

We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse $X\searrow Y$ of finite spaces induces a simplicial collapse $\k(X)\searrow \k(Y)$ of their associated simplicial complexes. Moreover, a simplicial collapse $K\searrow L$ induces a collapse $\x(K)\searrow \x(L)$ of the associated finite spaces. This establishes a one-to-one correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finite spaces. We also prove a similar result for maps: We give a complete characterization of the class of maps between finite spaces which induce simple homotopy equivalences between the associated polyhedra. Furthermore, this class describes all maps coming from simple homotopy equivalences at the level of complexes. The advantage of this theory is that the elementary move of finite spaces is much simpler than the elementary move of simplicial complexes: It consists of removing (or adding) just a single point of the space. Following the referee's suggestions, we have considerably rewritten sections 2, 3, and 4 of the paper. Although the mathematics involved is essentially the same in both versions, the exposition and the presentation of the results have substantially changed. As the referee pointed out, the exposition in the previous version was unsatisfactory, the texing was somewhat awkward and the organization of the article made for a good deal of repetition. We have corrected all these problems in this revised version. We are really grateful to the referee for his useful comments. * 1.5 Revision Notes