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Abstract

This thesis studies the geometric properties related to certain transversality statements on singular spaces, in a purely topological setting. These enter in the main part -- the construction of a generalized homology theory realized via bordism of such singular spaces -- through the inverse of the excision-isomorphism, the most difficult aspect of that problem. The relevancy of this homology theory is due to the unification of both, possessing a geometric description, establishing geometric fundamental-classes, and at the same time being well-suited to study inherently topological phenomena, like homeomorphism-invariance of said fundamental-classes, even in the absence of pl-structures. As an application, the invariance of Goresky--MacPherson L-classes under certain homeomorphisms is demonstrated.