Topological and Piecewise Linear Pseudoisotopy Functors

Abstract

We construct two functors <strong><em>P</em></strong><em>^strict</em>_∂:Top →Top and <strong>&Popf;</strong>^{<em>strict</em>}_∂:Top → Spectra such that for a compact manifold <em>M</em> the space <strong><em>P</em></strong>^{<em>strict</em>}_∂(<em>M</em>) has the homotopy type of the stable topological pseudoisotopy space of <em>M</em> and <strong>&Popf;</strong>^{<em>strict</em>}_∂(<em>M</em>) has the homotopy type of the topological pseudoisotopy spectrum of <em>M</em>. Both functors also induce homotopy functors that agree with the homotopy functor defined by Hatcher in 1978. The main idea of the construction is to build a homotopy coherent diagram out of induced maps as defined by Hatcher, then strictify the diagram and finally use a left Kan extension to extend the domain of the functor to the whole category Top of topological spaces. Our construction generalizes to the piecewise linear category and also yields piecewise linear versions of the two functors. <br /> The functor <strong><em>P</em></strong>^{<em>strict</em>}_∂ was already used in the work of other authors, although no complete construction of it existed prior to this work. We aim to close this gap in the literature.