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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Homotopy groups of the space of self-homotopy-equivalences


Author: Darryl McCullough
Journal: Trans. Amer. Math. Soc. 264 (1981), 151-163
MSC: Primary 55P10; Secondary 55N25, 55S35, 55S37, 57N65, 57T99
MathSciNet review: 597873
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Abstract: Let $ M$ be a connected sum of $ r$ closed aspherical manifolds of dimension $ n \geqslant 3$, and let $ EM$ denote the space of self-homotopy-equivalences of $ M$, with basepoint the identity map of $ M$. Using obstruction theory, we calculate $ {\pi _q}(EM)$ for $ 1 \leqslant q \leqslant n - 3$ and show that $ {\pi _{n - 1}}(EM)$ is not finitely-generated. As an application, for the case $ n = 3$ and $ r \geqslant 3$ we show that infinitely many generators of $ {\pi _1}(E{M^3},{\text{i}}{{\text{d}}_M})$ can be realized by isotopies, to conclude that $ {\pi _1}({\text{Homeo}}({M^3}),{\text{i}}{{\text{d}}_M})$ is not finitely-generated.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0597873-1
PII: S 0002-9947(1981)0597873-1
Keywords: Self-homotopy-equivalence, homeomorphism group, obstruction theory, aspherical manifold, $ 3$-manifold, isotopy
Article copyright: © Copyright 1981 American Mathematical Society