Homotopy classification of some four-dimensional manifolds

In this paper there is proved a generalization of the results of Whitehead and Pontryagin on the homotopy classification of closed, simply connected four-manifolds. Let W and M be compact four-dimensional simply connected oriented four-manifolds. By q_w is denoted the intersection index on the group H₂(W).Basic Result. THEOREM (Extension). Let the groups H₁(W)and H₁(M) be finite and suppose given a homotopy equivalence fWM. In order that f can be extended to a homotopy equivalence (W,W)(M,M), it is necessary and sufficient that there should exist an isomorphism , such that the diagram is commutative and *q_m=q_m.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 66, pp. 164–171, 1976.     

Homotopy classification of graded Picard categories

Certain low-dimensional symmetric cohomology groups of G-modules, for any given group G, are computed as the cohomology of an explicit cochain complex. This result is used to establish natural one-to-one correspondences between elements of the 3rd symmetric cohomology groups of G-modules, G-equivariant pointed 2-connected homotopy 4-types, and equivalence classes of G-graded Picard categories. The simplicial nerve of a G-graded Picard category is also constructed and studied.     

Homotopy classification of projective convex sets

A subset of projective space is called convex if its intersection with every line is connected. The complement of a projective convex set is again convex. We prove that for any projective convex set there exists a pair of complementary projective subspaces, one contained in the convex set and the other in its complement. This yields their classification up to homotopy.     

On domains with regular sections

We prove the generalized convexity of domains satisfying the condition of acyclicity of their sections by a certain continuously parametrized family of two-dimensional planes. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1420–1423, October, 2005.     

Homotopy classification of braided graded categorical groups

For any group G, a certain cohomology theory of G-modules is developed. This cohomology arises from the homotopy theory of G-spaces and it is called the “abelian cohomology of G-modules”. Then, as the main results of this paper, natural one-to-one correspondences between elements of the 3^rd cohomology groups of G-modules, G-equivariant pointed simply-connected homotopy 3-types and equivalence classes of braided G-graded categorical groups are established. The relationship among all these objects with equivariant quadratic functions between G-modules is also discussed.     

Homotopy classification of contact foliations on open contact manifolds

We give a homotopy classification of foliations on open contact manifolds whose leaves are contact submanifolds of the ambient space. The result is an extension of Haefliger’s classification of foliations on open manifold in the contact setting. While proving the main theorem, we also prove a result on equidimensional isocontact immersions on open contact manifolds.     

Remarks on the classification of quasitoric manifolds up to equivariant homeomorphism

We give three sufficient criteria for two quasitoric manifolds M, M′ to be (weakly) equivariantly homeomorphic. We apply these criteria to count the weakly equivariant homeomorphism types of quasitoric manifolds with a given cohomology ring.     

Homotopy classification of actions of finite groups on aspherical spaces

In this paper, actions of finite groups on aspherical spaces are studied. A complete homotopy classification of free actions of finite groups on aspherical spaces is obtained, including some results in nonfree actions. For example, a relation between cohomologies of finite groups and the lattice structure of their subgroups is obtained by this method; this relation is formulated in terms of spectral sequences.