topology (point-set topology, point-free topology)
see also 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
(also nonabelian homological algebra)
The singular homology of a topological space $X$ is the simplicial homology of its singular simplicial complex:
a singular $n$-chain on $X$ is a formal linear combination of singular simplices $\sigma : \Delta^n \to X$, and a singular $n$-cycle is such a chain such that its oriented boundary in $X$ vanishes. Two singular chains are homologous if they differ by a boudary. The singular homology of $X$ in degree $n$ is the group of $n$-cycles modulo those that are boundaries.
Singular homology of a topological space conincide with its ordinary homology as defined more abstractly (see at generalized homology theory).
(Here “singular” refers to the contrast with cellular homology, referring to the fact that a simplex $\Delta_{top} \to X$ in the singular simplicial complex is not required to be a topological embedding, but may be a “singular map”, such as for instance a constant function.)
Let $X \in$ Top be topological space. Write $Sing X \in$ sSet for its singular simplicial complex.
For $n \in \mathbb{N}$, a singular $n$-chain on $X$ is an element in the free abelian group $\mathbb{Z}[(Sing X)_n]$:
a formal linear combinations of singular simplices in $X$.
These are the chains on a simplicial set on $Sing X$.
The groups of singular chains combine to the simplicial abelian group $\mathbb{Z}[Sing X] \in Ab^{\Delta^{op}}$.
The alternating face map complex
is the singular complex of $X$.
Its chain homology is the ordinary singular homology of $X$.
One usually writes $H_n(X, \mathbb{Z})$ or just $H_n(X)$ for the singular homology of $X$ in degree $n$. See also at ordinary homology.
So we have
where the differentials are defined on basis elements $\sigma \in (Sing X)_n$ by
(with $d_i$ the $i$ simplicial face map) and then extended linearly.
(One may change the global signs and obtain a quasi-isomorphic complex, in particular with the same homology groups.)
This means that a singular chain is a cycle if the formal linear combination of the oriented boundaries of all its constituent singular simplices sums to 0. See the basic examples below
More generally, for $R$ any unital ring one can form the degreewise free module $R[Sing X]$ over $R$. The corresponding homology is the singular homology with coefficients in $R$, denoted $H_n(X,R)$.
Given a continuous map $f : X \to Y$ between topological spaces, and given $n \in \mathbb{N}$, every singular $n$-simplex $\sigma : \Delta^n \to X$ in $X$ is sent to a singular $n$-simplex
in $Y$. This is called the push-forward of $\sigma$ along $f$. Accordingly there is a push-forward map on groups of singular chains
These push-forward maps make all diagrams of the form
commute. In other words, push-forward along $f$ constitutes a chain map
It is in fact evident that push-forward yields a functor of singular simplicial complexes
From this the statement follows since $\mathbb{Z}[-] : sSet \to sAb$ is a functor.
Accordingly we have:
Sending a topological space to its singular chain complex $C_\bullet(X)$, def. 2, and a continuous map to its push-forward chain map, prop. 1, constitutes a functor
from the category Top to the category of chain complexes.
In particular for each $n \in \mathbb{N}$ singular homology extends to a functor
Let $X$ be a topological space. Let $\sigma^1 : \Delta^1 \to X$ be a singular 1-simplex, regarded as a 1-chain
Then its boundary $\partial \sigma \in H_0(X)$ is
or graphically (using notation as for orientals)
Let $\sigma^2 : \Delta^2 \to X$ be a singular 2-chain. The boundary is
Hence the boundary of the boundary is
For more illustrations see for instance (Ghrist, (4.5)).
For all $n \in \mathbb{N}$ the reduced singular homology of the $n$-sphere $S^n$ is
The $n$-sphere may be realized as the pushout
which is the $n$-ball with its boundary $(n-1)$-sphere identified with the point. The inclusion $S^{n-1} \hookrightarrow D^n$ is a “good pair” in the sense of def. 5, and so the long exact sequence from prop. 7 yields a long exact sequence
Since the disks are all contractible topological spaces we have $H_k(D^n) \simeq 0$ for all $k,n$ by this example at reduced homology. This means that in the above long exact sequence all the morphisms
are isomorphisms, for all $k \in \mathbb{N}$. Since
(by this example at reduced homology) the statement follows by induction on $n$.
Singular homology is homotopy invariant:
If $f : X \to Y$ is a continuous map between topological spaces which is a homotopy equivalence, then the induced morphism on singular homology groups
is an isomorphism.
In other words: the singular chain functor of prop. 2 sends weak homotopy equivalences to quasi-isomorphisms.
A proof (via CW approximations) is spelled out for instance in (Hatcher, prop. 4.21).
The singular homology groups of a topologial space serve to some extent as an approximation to the homotopy groups of that space.
(Hurewicz homomorphism)
For $(X,x)$ a pointed topological space, the Hurewicz homomorphism is the function
from the $k$th homotopy group of $(X,x)$ to the $k$th singular homology group defined by sending
a representative singular $k$-sphere $f$ in $X$ to the push-forward along $f$ of the fundamental class $[S_k] \in H_k(S^k) \simeq \mathbb{Z}$.
For $X$ a topological space the Hurewicz homomorphism in degree 0 exhibits an isomorphism between the free abelian group $\mathbb{Z}[\pi_0(X)]$ on the set of connected components of $X$ and the degree-0 singular homlogy:
Since a homotopy group in positive degree depends on the homotopy type of the connected component of the base point, while the singular homology does not depend on a basepoint, it is interesting to compare these groups only for the case that $X$ is connected.
For $X$ a connected topological space the Hurewicz homomorphism in degree 1
is surjective. Its kernel is the commutator subgroup of $\pi_1(X,x)$. Therefore it induces an isomorphism from the abelianization $\pi_1(X,x)^{ab} \coloneqq \pi_1(X,x)/[\pi_1,\pi_1]$:
For higher connected $X$ we have the
If $X$ is (n-1)-connected for $n \geq 2$ then
is an isomorphism.
This is known as the Hurewicz theorem.
For the present purpose one makes the following definition.
A topological subspace inclusion $A \hookrightarrow X$ in Top is called a good pair if
$A$ has a neighbourhood in $X$ of which it is a deformation retract.
Write $X/A$ for the cokernel of the inclusion, hence for the pushout
in Top.
If $A \hookrightarrow X$ is a good pair, def. 5, then the singular homology of $X/A$ coincides with the relative homology of $X$ relative to $A$. In particular, therefore, it fits into a long exact sequence of the form
For instance (Hatcher, theorem 2.13).
Singular homology computes the generalized homology with coefficients in the Eilenberg-MacLane spectrum $H \mathbb{Z}$ or $H R$.
The dual notion is that of singular cohomology.
The analogous notion in algebraic geometry is given by Chow groups.
Lecture notes include
Textbook discussion in the context of homological algebra is around Application 1.1.4 of
and in the context of algebraic topology in chapter 2.1 of
and chapter 4 of
Discussion in the context of computing homotopy groups is in
Lecture notes include
See also