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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the $ 2$-realizability of $ 2$-types

Author: Micheal N. Dyer
Journal: Trans. Amer. Math. Soc. 204 (1975), 229-243
MSC: Primary 55D15
MathSciNet review: 0380783
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Abstract: A $ 2$-type is a triple $ (\pi ,{\pi _2},k)$, where $ \pi $ is a group, $ {\pi _2}$ a $ \pi $-module and $ k \in {H^3}(\pi ,{\pi _2})$. The following question is studied: When is a $ 2$-type $ (\pi ,{\pi _2},k)$ realizable by $ 2$-dimensional CW-complex $ X$ such that the $ 2$-type $ ({\pi _1}X,{\pi _2}X,k(X))$ is equivalent to $ (\pi ,{\pi _2},k)$? A long list of necessary conditions is given (2.2). One necessary and sufficient condition (3.1) is proved, provided $ \pi $ has the property that stably free, finitely generated $ \pi $-modules are free. ``Stable'' $ 2$-realizability is characterized (4.1) in terms of the Wall invariant of [15]. Finally, techniques of [5] are used to extend C. T. C. Wall's Theorem F of [15] to a space $ X$ which is dominated by a finite CW-complex of dimension 2, provided $ {\pi _1}X$ is finite cyclic. Under these conditions $ X$ has the homotopy type of a finite $ 2$-complex if and only if the Wall invariant vanishes.

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Article copyright: © Copyright 1975 American Mathematical Society