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.