group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Given that the (degree-shifted, reduced) topological K-theory group of a topological space $X$ can be computed as a colimit over sets of homotopy classes of maps (cf. at stable unitary group)
one may think of this as a stabilization of generalized nonabelian cohomology theories (in the sense of FSS23, §2) $U_n(X)$ whose classifying spaces are the unitary groups $U(n)$. For this reason, we may call the groups $[X,U(n)] \,\coloneqq\, \pi_0 Map\big(X,U(n)\big)$ the unstable $\tilde{K}^1$-theory groups of $X$, at stage $n$.
Let $X$ be a topological space (CW-complex), and $U(n)$ the unitary group in dimension $n$. The $n$-unstable $\tilde{K}^1$-theory group $U_n(X)$ of $X$ is defined as the homotopy classes of maps, hence the connected components of the space of maps from $X$ to (the underlying topological space of) $U(n)$:
From this definition one can see that, if $X$ is a finite-dimensional CW complex then $U_n(X) = \tilde{K}^1 (X)$ for sufficiently large $n$.
The following properties are proven in Hamanaka and Kono 2003:
(Theorem 1.1 of op.cit.)
Let $\text{dim}(X)\leq 2n$. Then there exists an exact sequence of the form:
or, put differently, defining $N_n (X) \coloneqq coker(\Theta)$ (see Section 3 of op.cit. for the definition of $\Theta$):
In particular, Prop. shows that in general $U_n(X)$ is neither abelian nor does it inject into $\tilde{K}^1 (X)$.
(Theorem 1.2 of op.cit.)
For $dim(X)\leq 2n$, the cokernel group $N_n(X)$ is a finite abelian group where the order of any element divides $n!$.
An example where not only do the unstable and stable K theory groups not coincide but the latter actually vanishes is provided by the even-dimensional spheres.
(Lemma 4.1 of Hamanaka 2003)
For $n\geq 3$, the $n$ unstable K-theory group of the $2n$-dimensional sphere $S^{2n}$ is
whereas $\tilde{K}^1 (S^{2n} ) =0$.
For the notion of unstable algebraic K-theory, see Jansen 2024.
Hiroaki Hamanaka, Akira Kono: On $[X, U(n)]$ when $\text{dim}(X)$ is $2n$, Journal of Mathematics of Kyoto University 43 2 (2003) 333-348 [doi:10.1215/kjm/1250283730]
Hiroaki Hamanaka, Akira Kono: An application of unstable K-theory, Journal of Mathematics of Kyoto University 44 2 (2004) 451-456 [doi:10.1215/kjm/1250283560]
Hiroaki Hamanaka: Adams $e$-invariant, Toda bracket and $[X, U (n)]$, Journal of Mathematics of Kyoto University 43 4 (2003) 815-827 [doi:10.1215/kjm/1250281737]
On unstable algebraic K-theory:
Last revised on June 15, 2024 at 17:43:59. See the history of this page for a list of all contributions to it.