homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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 topological space/simplicial set/homotopy type/infinity-groupoid $X$ is nilpotent if
its fundamental group is a nilpotent group
the canonical action of $\pi_1(X)$ on all the higher homotopy groups $\pi_{n \geq 2}(X)$ is nilpotent.
The first condition means that $\pi_1(X)$ is isomorphic to an iterated central extension of abelian groups.
The second condition means that each $\pi_{n \geq 2}(X)$ admits a sequence of subgroups
such that for all $i$
$G_{i+1} \hookrightarrow G_i$ is a normal subgroup;
the quotient $G_i/G_{i+1}$ is an abelian group;
each $G_i$ is closed under the action of $\pi_1(X)$;
the induced action on $G_i/G_{i+1}$ is trivial.
This implies that given any element $a \in \pi_{n \geq 2}(X)$, then after acting on it at most $k$ times with elements from $\pi_1(X)$ the result is zero.
Nilpotency is involved in sufficient conditions for many important constructions in (stable) homotopy theory, see for instance at
Review includes
See also
Last revised on April 19, 2018 at 03:11:26. See the history of this page for a list of all contributions to it.