This talk will present certain homological invariants attached to a stratified space \(X\), called the Whitney-deRham cohomology. This cohomology is defined as the cohomology of a chain complex, \(\Omega^{ * }_{\mathrm{ W } } \left( X \right)\), associated to \(X\) in a somewhat ad-hoc way. The main result is to show that though the definition of \(\Omega^{ * }_{\mathrm{ W } } \left( X \right)\) depends on several choices, when certain conditions are imposed on \(X\), the Whitney-deRham cohomology only depends on the homotopy type of \(X\). This is achieved by showing that \(\Omega^{ * }_{\mathrm{ W } } \left( X \right)\) can be realized as a fine complex of sheaves which is a resolution of the locally constant sheaf on \(X\). An application of this work is in the area of homotopy theory. One can canonically define a commutative differential graded algebra(CDGA), \(A_{PL}\left(X\right)\), on \(X\) in such a way that any CDGA which is quasi-isomorphic to it determines the real homotopy type of \(X\). It will be shown that the complex \(\Omega^{ * }_{\mathrm{ W } } \left( X \right)\) is quasi-isomorphic to \(A_{PL}\left(X\right)\), and thus determines the real homotopy type of \(X\).
The real homotopy type of singular spaces via the Whitney-deRham Complex
The real homotopy type of singular spaces via the Whitney-deRham Complex