We show a Whitney approximation theorem for continuous map from manifold to smooth CW complex, which enables us that topological complex is homotopy equivalent in category of spaces. It also shown that, any open covering there exists partition unity subordinate the covering. In addition, we observe are enough many functions on complex.

We describe CW complexes for complex projective algebraic surfaces in the context of practical computation of topological invariants. CW COMPLEXES FOR COMPLEX ALGEBRAIC SURFACES

LetG be a locally compact topological group and EG its universal space for the family of compact subgroups. We give criteria for this space to be G-homotopy equivalent to a d-dimensional G-CW -complex, a finite G-CW -complex or a G-CW -complex of finite type. Essentially we reduce these questions to discrete groups, and to the homological algebra of the orbit category of discrete groups with re...

We use algebraic topology to study the stochastic motion of cellular cycles in a finite CW complex. Inspired by statistical mechanics, we introduce homological observable called average current. The latter measures flux probability process. In low temperature, adiabatic limit, prove that current fractionally quantizes, which denominators are combinatorial invariants

The C* -algebra extensions of a topological space can be made into an abelian group which is naturally equivalent to the Khomology group of odd dimension [1] which has a close relation with index theory and is one of the starting points of KK theory [8]. The Cp -smoothness of an extension of a manifold was introduced in [3, 4], where Cp denotes the Schatten-von Neumann p-class [5]. We generaliz...