Homotopy classification of some four-dimensional manifolds

O. A. Ivanov

Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part II, Tome 66 (1976), pp. 164-171

Résumé

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 $H_1(\partial W)$ and $H_1(\partial M)$ be finite and suppose given a homotopy equivalence $f:\partial W\to\partial M$. In order that $f$ can be extended to a homotopy equivalence $(W,\partial W)\to(M,\partial M)$, it is necessary and sufficient that there should exist an isomorphism $\Xi$, such that the diagram $$ \begin{array}{ccc} H_2(W,\partial W) \overset {\partial}\longrightarrow H_1(\partial W) \\ \downarrow\Xi \downarrow f*\\ H_2(M,\partial M) \overset {\partial}\longrightarrow H_1(\partial W) \end{array} $$ is commutative and $\Xi^*q_m=q_w$.