The Real Homotopy Type of Singular Spaces via The Whitney-deRham Complex.

摘要

This thesis studies certain homological invariants attached to a stratied space X, called the Whitney-de Rham cohomology. This cohomology is defined as the cohomology of a commutative dierential graded algebra (CDGA), W (X), associated to X in a way that depends on several choices. The main result is to show that when X is a semi-analytic set, the Whitney-de Rham cohomology only depends on the homotopy type of X. This is achieved by showing that W (X) is in fact weekly equivalent to the singular cochain complex on X. The main results is as follows: Theorem 1: Let X be a semi-analytic subset of n-dimensional Euclidean space, then the complex of sheaves of germs of Whitney-de Rham differential forms on X is a fine resolution of the locally constant sheaf on X. Thus the Whitney-de Rham cohomology of X is isomorphic to the real singular cohomology of X. An application of this is in the area of homotopy theory. If one has a CDGA, A, that is quasi-isomorphic to a specific CDGA, APL (X), which is functorialy defined on X, then one can determine the real homotopy type of X directly from A. This yields the following result: Theorem 2: Let X be a semi-analytic set such that X is simply connected and homologically of finite type, then the Whitney-de Rham complex, W (X), determines the real homotopy type of X..