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 Mat. Sb. (N.S.), 1969, Volume 79(121), Number 3(7), Pages 307–356 (Mi msb3591)

Local contractibility of the group of homeomorphisms of a manifold

A. V. Černavskiĭ

Abstract: In this paper the group of homeomorphisms of an arbitrary topological manifold is considered, with either the compact-open, uniform (relative to a fixed metric), or majorant topology. In the latter topology, a basis of neighborhoods of the identity is given by the strictly positive functions on the manifold, a homeomorphism being in the neighborhood determined by such a function if it moves each point less than the value of this function at the point. The main result of the paper is the proof of the local contractibility of the group of homeomorphisms in the majorant topology. Examples are easily constructed to show that this assertion is false for the other two topologies for open manifolds. In the case of a compact manifold the three topologies coincide. In conclusion a number of corollaries are given; for example, if a homeomorphism of a manifold can be approximated by stable homeomorphisms then it is itself stable.
Figures: 4.
Bibliography: 14 titles.

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English version:
Mathematics of the USSR-Sbornik, 1969, 8:3, 287–333

Bibliographic databases:

UDC: 513.836
MSC: 58D05, 57S05

Citation: A. V. Černavskiǐ, “Local contractibility of the group of homeomorphisms of a manifold”, Mat. Sb. (N.S.), 79(121):3(7) (1969), 307–356; Math. USSR-Sb., 8:3 (1969), 287–333

Citation in format AMSBIB
\Bibitem{Che69} \by A.~V.~{\v C}ernavski{\v\i} \paper Local contractibility of the group of homeomorphisms of a~manifold \jour Mat. Sb. (N.S.) \yr 1969 \vol 79(121) \issue 3(7) \pages 307--356 \mathnet{http://mi.mathnet.ru/msb3591} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=259925} \zmath{https://zbmath.org/?q=an:0184.26801|0193.51501} \transl \jour Math. USSR-Sb. \yr 1969 \vol 8 \issue 3 \pages 287--333 \crossref{https://doi.org/10.1070/SM1969v008n03ABEH001121} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. F.C. Tinsley, David G. Wright, “Some contractible open manifolds and coverings of manifolds in dimension three”, Topology and its Applications, 77:3 (1997), 291
2. P. M. Akhmet'ev, “Embedding of compacta, stable homotopy groups of spheres, and singularity theory”, Russian Math. Surveys, 55:3 (2000), 405–462
3. Melikhov, SA, “On maps with unstable singularities”, Topology and Its Applications, 120:1–2 (2002), 105
4. S. A. Melikhov, “Isotopic and continuous realizability of maps in the metastable range”, Sb. Math., 195:7 (2004), 983–1016
5. A. V. Chernavskii, “On the work of L. V. Keldysh and her seminar”, Russian Math. Surveys, 60:4 (2005), 589–614
6. A. V. Chernavskii, “Local Contractibility of the Homeomorphism Group of $\mathbb R^n$”, Proc. Steklov Inst. Math., 263 (2008), 189–203
7. PAUL A. SCHWEITZER, S. J, “Normal subgroups of diffeomorphism and homeomorphism groups of ℝ n and other open manifolds”, Ergod. Th. Dynam. Sys, 2011, 1
8. A. N. Dranishnikov, “On some problems related to the Hilbert-Smith conjecture”, Sb. Math., 207:11 (2016), 1562–1581
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