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On the structure of the second homotopy group of the union of two spaces. (Russian) Zbl 0584.55011

The Hurewicz theorem in dimension 1 (that \(H_ 1(X)\) is the abelianization of \(\pi_ 1(X))\), and van Kampen’s theorem, are effective tools for the study of \(\pi_ 1\) of rather general spaces. Some work along the same lines has been done for \(\pi_ 2\), for example S. J. Lomonaco jun. [Pac. J. Math. 95, 349-390 (1981; Zbl 0483.57012)]. Working with connected finite cellular spaces the author considers triads \((X,X_+,X_-)\) for which \(X=X_+\cup X_-\), with \(X_ 0=X_+\cap X_-\) a finite connected complex.
His principal theorem describes \(\pi_ 2(X)\) in the case where the maps \(X_ 0\subseteq X_+\) and \(X_ 0\subseteq X_-\) induce epimorphisms on the fundamental groups and \(\pi_ 2(X_+)=\pi_ 2(X_-)=0\). If \(N_{\pm}\) are the kernels of the maps which are induced by the above inclusions on the fundamental groups, then \(\pi_ 2(X)\) is isomorphic (with \(\pi_ 1\) action) to \((N_+\cap N_-)/[N_+,N_-]\). The author also connects this work with problems about knots in \({\mathbb{R}}^ 4\).
Reviewer: D.Kahn

MSC:

55Q05 Homotopy groups, general; sets of homotopy classes
55Q52 Homotopy groups of special spaces
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)

Citations:

Zbl 0483.57012
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