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
(see also Chern-Weil theory, parameterized homotopy theory)
Generally, given a covering space $E \to X$, a deck transformation is a bundle-automorphism, hence a continuous map $\delta \,\colon\, E \xrightarrow{\;\sim\;} E$ fixing the base $X$.
Assuming without real restriction that $X$ is connected, then $E \to X$ is a principal bundle with discrete fibers and the deck transformation determines and is determined by the element $\delta_x \,\colon\, E_x \xrightarrow{\;\sim\;} E_x$ of the permutation group $Aut(E_x)$ of its fiber $E_x$ over any fixed basepoint $x \in X$.
When the fundamental theorem of covering spaces applies (i.e. when $X$ is locally path-connected and semi-locally simply-connected topological space) then the fundamental group $\pi_1(X,x)$ has a canonical action on this fiber, by a group homomorphism $\pi_1(X,x) \xrightarrow{\;} Aut(E_x)$, and often it is specifically this $\pi_1$-action which is referred to as being by deck transformations.
In particular, in the case that $E \to X$ is the universal cover, then $E_x \,\simeq\, \pi_1(X)$ and $E \to X$ is a $\pi_1(X)$-principal bundle with the fundamental group $\pi_1(X)$ being identified with the group of deck transformations.
A deck transformation or cover automorphism is an automorphism of a covering space relative to the base space.
i.e. if $p\colon E\to X$ is a cover then a cover automorphism $f\in deck(p)=\{f|f\in Aut(E), p\circ f=p\}\subseteq Aut(E)$ is an automorphism of $E$ such that $p$ is invariant under composition with $f$.
Any of the references at covering space.
See also:
Wikipedia, Deck transformation
MathWorld, Deck transformation
Last revised on August 18, 2022 at 04:06:50. See the history of this page for a list of all contributions to it.