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
A vectorial bundle (Gomi 08) is a $\mathbb{Z}_2$-graded vector bundle $E$ of finite rank, equipped with an odd endomorphism $h \;\colon\; E \to E$. The homomorphisms of vectorial bundles are such that the endomorphism $h$ acts like canceling parts of the even and odd degree of $E$ against each other.
This way vectorial bundles lend themselves to the description of topological K-theory. In particular, they allow a geometric model for twisted K-theory.
For $X$ a topological space, the category $VectrBund(X)$ of vectorial bundles on $X$ has
as objects $(E \stackrel{h}{\to} E)$ finite rank Hermitean $\mathbb{Z}_2$-graded vector bundles $E\to X$ equipped with a self-adjoint endomorphism $h$ of odd degree. In matrix calculus
as morphisms $\phi : (E,h) \to (E',h)$ equivalence classes of morphisms $\phi \colon E \to E'$ of vector bundles such that
where two such maps are regarded as equivalent, $\phi \sim \phi'$, already if they coincide on the kernel of $h^2_x$ for each point $x$.
In particular, we have the following two important special cases:
the case that $h = 0$ – in this case all eigenvalues of all $h_x^2$ are zero. and hence maps $\phi, \phi' : (E,0) \to (E',0)$ represent the same morphism precisely if they are actually equal as morphisms $\phi, \phi' : E \to E'$ of vector bundles.
(Notice that there is only the 0-morphism $(E,0) \to (E',h')$ for $h' \neq 0$.)
This yields a canonical inclusion
by sending $E \mapsto (E \stackrel{0}{\to} E)$.
the case that $E = \left( \array{V \\ V}\right)$ and $h = \left( \array{ 0 & Id \\ Id & 0 } \right)$
Here $E_x|_{\lt \mu \lt 1} = 0$ and hence two morphisms $\phi, \phi' : (E,h) \to (E',h')$ are identified already if they agree on the 0-vector. In other words, all morphisms out of such $(E,h)$ are identified. In particular they are all equal to the 0-morphism to $(0,0)$. Therefore the bundles of this form represent the 0-element.
Definition
Say two vectorial bundles $(E,h)$, $(E',h')$ on $X$ are concordant if there is a vectorial bundle on $X \times [0,1]$ which restricts to them at either end, respectively.
Let $(E,h)^{\vee} =$ be the degree-reversed bundle to $(E,h)$.
Lemma
There is a concordance
The definition of vectorial bundles is due to Furuta. It is recalled and applied to the study of K-theory and twisted K-theory in
Last revised on November 2, 2023 at 19:56:32. See the history of this page for a list of all contributions to it.