Classifying homeomorphism groups of infinite graphs

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

“…In this paper we first show that the group H c (M ) is locally contractible in any dimension n. As mentioned above, the conjecture above has been proved in the case n = 1, see [5]. Here we solve the conjecture affirmatively in the case n = 2.…”

“…Moreover, it is proved in [5] that if M is a non-compact separable graph then H c (M ) is an (l 2 ×R ∞ )-manifold.…”

The diffeomorphism groups of the real line are pairwise bihomeomorphic

“…In this case the triple (D 0 (R), D 0 + (R), D 0 c (R)) is denoted by (H(R), H + (R), H c (R)). In [3] it was proved that the pair (H + (R), H c (R)) is weakly and strongly homeomorphic to the pair (Π ω l 2 , Σ ω l 2 ) (surprisingly, but we do not know if these two pairs are bihomeomorphic! )…”

“…By [3] the pairs of homeomorphism groups (H + (R), H c (R)) and (H(R), H c (R)) endowed with the Whitney topology are homeomorphic to the pair ( ω l 2 , ⊡ ω l 2 ). This fact combined with Theorem 1.2 implies On the other hand, the pair of homeomorphism groups (H + (R), H c (R)) endowed with the compactopen topology is homeomorphic to the pair (Π ω l 2 , Σ ω l 2 ), see [3]. Combining this fact with Theorem 1.2 we get another…”

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

“…For non-compact graphs a counterpart of Theorem 2 was proved in [3]. Next, we calculate the cardinality of the mapping class group M c (M ) = H c (M )/H 0 (M ) of a connected surface M .…”