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
Generally, given some kind of space equipped with the action of a group, the locus of fixed points of the action may form a suitable sub-space: the fixed point space.
Specifically, given a topological group $G$ and a topological G-space, its fixed point space is the set of the set-theoretic fixed points of the $G$-action, equipped with the subspace topology.
For more see at topological G-space the section Change of groups and fixed loci.
The statement of Elmendorf's theorem is essentially that the equivariant homotopy theory of topological $G$-spaces is equivalently encoded in their systems of $H$-fixed point spaces, as $H$ varies over closed subgroups of $G$.
In equivariant stable homotopy theory the concept of fixed point spaces branches into various closely related, but different concepts:
categorical fixed point spectra?
In equivariant differential topology:
(existence of $G$-invariant tubular neighbourhoods)
Let $X$ be a smooth manifold, $G$ a Lie group and $\rho \;\colon\; G \times X \to X$ a proper action by diffeomorphisms.
If $\Sigma \overset{\iota}{\hookrightarrow} X$ is a closed smooth submanifold inside the $G$-fixed locus
then $\Sigma$ admits a $G$-invariant tubular neighbourhood $\Sigma \subset U \subset X$.
Moreover, any two choices of such $G$-invariant tubular neighbourhoods are $G$-equivariantly isotopic.
(Kankaanrinta 07, theorem 4.4, theorem 4.6)
(fixed loci of smooth proper actions are submanifolds)
Let $X$ be a smooth manifold, $G$ a Lie group and $\rho \;\colon\; G \times X \to X$ a proper action by diffeomorphisms.
Then the $G$-fixed locus $X^G \hookrightarrow X$ is a smooth submanifold.
(see also this MO discussion)
Let $x \in X^G \subset X$ be any fixed point. Since this is in particular a closed invariant submanifold, Prop. applies and shows that an open neighbourhood of $x$ in $X$ is $G$-equivariantly diffeomorphic to a linear representation $V \in RO(G)$. The fixed locus $V^G \subset V$ of that is hence diffeomorphic to an open neighbourhood of $x$ in $\Sigma$.
Without the assumption of proper action in Prop. the conclusion generally fails. See this MO comment for a counter-example.
For $G$ a topological group, consider the category of TopologicalGSpaces.
For $H \subset G$ any subgroup, consider
the coset space $G/H \in Topological G Spaces$;
the Weyl group $W_H(G) \coloneqq N_G(H)/H \in TopologicalGroups$.
Observing that for $X \in Topological G Spaces$ the $H$-fixed locus $X^H$ inherits a canonical action of $N(H)/H$, we have a functor
Notice that $G$ acts canonically on the left of $G/H$, while $N(H)/H$ still acts from the right (both by group multiplication on representatives):
Therefore there exists a functor in the other direction:
(passage to fixed loci is a right adjoint)
These are adjoint functors, with the $H$-fixed locus functor (1) being the right adjoint:
To see the hom-isomorphism characterizing this adjunction, consider for $X \in Topological N(H)/H Spaces$ and $Y \in Topological G Spaces$ a $G$-equivariant continuous function
This restricts to an $N(H)$-equivariant function on the $N(H)$-topological subspace
Since $Y$ is a fixed locus for $H \subset N(H)$, by equivariance this restriction has to factor through the $H$-fixed locus $X^H$ of $X$:
But given that and since every other point of $G/H \times_{N(H)/H} Y$ is an image under the $G$-action of a point in $Y \subset G/H \times_{N(H)/H}$, this restriction $\tilde f$ already determines $f$ uniquely.
Since this construction is manifestly natural in $Y$ and $X$, we have a natural bijection $f \leftrightarrow \tilde f$, which establishes the hom-isomorphism for the pair of adjoint functors in (2).
The adjunction (2) factors as:
Here the functor on the top right, $N(H)/H \times_{N(H)/H} (-)$, is the identity on the underlying topological spaces, but extends the action from $N(H)/H$ to $N(H)$, namely through the projection homomorphims $N(H) \to N(H)/H$.
For more on this see at Topological G-space – Fixed loci with residual Weyl gorup action:
Last revised on September 22, 2024 at 06:19:50. See the history of this page for a list of all contributions to it.