Diffeomorphism groups of non-compact manifolds endowed with the Whitney<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mo>∞</mml:mo></mml:mrow></mml:msup></mml:math>-topology

Банах, Тарас; Radul, Taras

doi:10.1016/j.topol.2014.08.015

Cited by 4 publications

(11 citation statements)

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“…In particular, the Lie groups lim −→ G n (as in [21]) are complete for each strict direct sequence G 1 ⊆ G 2 ⊆ · · · of finite-dimensional Lie groups. 6 Diffeomorphism groups. For M a paracompact finite-dimensional smooth manifold, consider the group Diff c (M) of all C ∞ -diffeomorphisms φ : M → M with compact support (in the sense that φ(x) = x for x outside some compact set).…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…(2) 6 The Lie group structure on important examples of such groups (like GL ∞ (R) = lim −→ GL n (R) and SL ∞ (R) = lim −→ SL n (R)) was already constructed in [36]; cf. also [39].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…We mention that the topology on the Lie group Diff c (M ) coincides with the Whitney C ∞ -topology used in[6]; this is clear from the description of this topology in[34] (see also[32] for a detailed account).…”

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confidence: 96%

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Completeness of Infinite-dimensional Lie Groups in Their Left Uniformity

Glöckner¹

2019

Can. J. Math.-J. Can. Math.

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Abstract. We prove completeness for the main examples of in nite-dimensional Lie groups and some related topological groups. Consider a sequence G ⊆ G ⊆ ⋅ ⋅ ⋅ of topological groups G n such that G n is a subgroup of G n+ and the latter induces the given topology on G n , for each n ∈ N. Let G be the direct limit of the sequence in the category of topological groups. We show that G induces the given topology on each G n whenever ⋃ n∈N V V ⋅ ⋅ ⋅ V n is an identity neighbourhood in G for all identity neighbourhoods V n ⊆ G n . If, moreover, each G n is complete, then G is complete. We also show that the weak direct product ⊕ j∈J G j is complete for each family (G j ) j∈J of complete Lie groups G j . As a consequence, every strict direct limit G = ⋃ n∈N G n of nite-dimensional Lie groups is complete, as well as the di eomorphism group Di c (M) of a paracompact nite-dimensional smooth manifold M and the test function group C k c (M, H), for each k ∈ N ∪ {∞} and complete Lie group H modelled on a complete locally convex space.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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Completeness of Infinite-dimensional Lie Groups in Their Left Uniformity

Glöckner¹

2019

Can. J. Math.-J. Can. Math.

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“…For a surface M let H(M ) denote the homeomorphism group of M endowed with the Whitney topology. This topology is generated by the base consisting of the sets The local topological structure of the groups H 0 (M ; K) and H c (M ; K) was studied in [4] for the case when K is a subpolyhedron in a surface M . It was shown in [4] that the topological group H c (M ; K) is…”

Section: Introductionmentioning

confidence: 99%

“…Another result of [4] says that the topological group H c (M ; K) is locally contractible and hence the connected component H 0 (M ; K) is an open subgroup of H c (M ; K). Consequently, H c (M ; K) is homeomorphic to the product H 0 (M ; K) × M c (M ; K) of the connected group H 0 (M ; K) and the discrete quotient group M c (M ; K) = H c (M ; K)/H 0 (M ; K), which can be called the mapping class group of the pair (M, K).…”

Section: Introductionmentioning

confidence: 99%

On homeomorphism groups of non-compact surfaces, endowed with the Whitney topology

Банах

Mine

Sakai

et al. 2014

Topology and its Applications

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Detecting topological groups which are (locally) homeomorphic to LF-spaces

Банах

Mine

Repovš

et al. 2013

Topology and its Applications

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We prove that a topological group G is (locally) homeomorphic to an LF-space if G = n∈ω Gn for some increasing sequence of subgroups (Gn)n∈ω such that (1) for any neighborhoods Un ⊂ Gn, n ∈ ω, of the neutral element e ∈ Gn ⊂ G, the set(2) each group Gn is (locally) homeomorphic to a Hilbert space;(3) for every n ∈ N the quotient map Gn → Gn/G n−1 is a locally trivial bundle; (4) for infinitely many numbers n ∈ N each Z-point in the quotient space Gn/G n−1 = {xG n−1 : x ∈ Gn} is a strong Z-point.2010 Mathematics Subject Classification. 57N20; 46A13; 22A05.

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Diffeomorphism groups of non-compact manifolds endowed with the WhitneyC∞-topology

Cited by 4 publications

References 49 publications

Completeness of Infinite-dimensional Lie Groups in Their Left Uniformity

Completeness of Infinite-dimensional Lie Groups in Their Left Uniformity

On homeomorphism groups of non-compact surfaces, endowed with the Whitney topology

Detecting topological groups which are (locally) homeomorphic to LF-spaces

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