Vladimir Abramovich Rokhlin and algebraic topology

Abstract

The article considers the scientific heritage of V. A. Rokhlin in algebraic topology from the point of view of the modern development of mathematics and shows the influence of his results on the development of algebraic topology up to the present. The second part of the article contains new results with fairly detailed sketches of their proofs. There we introduce the notion of partially framed manifolds, which naturally arise in the study of the characteristic classes of vector bundles over the loop space Ω S U ( 2 ) = Ω S P ( 1 ) \Omega SU(2)=\Omega SP(1) . We obtain theorems on the divisibility of the signature of such manifolds as a result of calculations of characteristic classes with values in complex and quaternionic cobordism.