nLab
singular cohomology

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

The singular cohomology (also Betti cohomology) of a topological space XX is the cohomology in ∞Grpd of its fundamental ∞-groupoid Π(X)\Pi(X):

for nGrpd\mathcal{B}^n \mathbb{Z} \in \infty Grpd the Eilenberg-MacLane object with the group \mathbb{Z} in degree nn, the degree nn-singular cohomology of XX is

H n(X,):=π 0Grpd(Π(X), n). H^n(X,\mathbb{Z}) := \pi_0 \infty Grpd(\Pi(X), \mathcal{B}^n \mathbb{Z}) \,.

With Grpd\infty Grpd presented by the category sSet of simplicial sets, the fundamental \infty-groupoid Π(X)\Pi(X) is modeled by the Kan complex

Π(X)=SingX=Hom Top(Δ Top ,X), \Pi(X) = Sing X = Hom_{Top}(\Delta^\bullet_{Top}, X) \,,

the singular simplicial complex of XX.

The object n\mathcal{B}^n \mathbb{Z} is usefully modeled by the simplicial set

n=U(Ξ[n]) \mathcal{B}^n \mathbb{Z} = U (\Xi \mathbb{Z}[n])

which is

  • the underlying simplicial set under the forgetful functor

    (FU)sAbUFsSet (F \dashv U) sAb \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} sSet

    from abelian simplicial groups to simplicial sets;

  • of the abelian simplicial group Ξ[n]\Xi \mathbb{Z}[n] which is the image under the Dold-Kan correspondence

    sAbΞCh + sAb \stackrel{\overset{\Xi}{\leftarrow}}{\underset{}{\to}} Ch^+
  • of the chain complex

    [n]=(000) \mathbb{Z}[n] = (\cdots \to \mathbb{Z} \to 0 \to 0 \to \cdots \to 0)

    concentrated in degree nn.

So in this model we have

H n(X,)=π 0sSet(SingX,U(Ξ[n])). H^n(X,\mathbb{Z}) = \pi_0 sSet(Sing X, U(\Xi \mathbb{Z}[n])) \,.

A cocycle in this cohomology theory is a cochain on a simplicial set, on the singular complex SingXSing X.

Using the adjunction (FU)(F \dashv U) this is isomorphic to

π 0sAb(Ch n(X),Ξ[n]), \cdots \simeq \pi_0 sAb( Ch_n(X), \Xi \mathbb{Z}[n] ) \,,

where

F(SingX)=[SingX] F(Sing X) = \mathbb{Z}[Sing X]

is the free abelian simplicial group on the simplicial set SingXSing X: this is the simplicial abelian group of singular chains of XX. Its elements are formal sums of continuous maps Δ Top nX\Delta^n_{Top} \to X. In this form

π 0sAb([SingX],Ξ[n]). \cdots \simeq \pi_0 sAb( \mathbb{Z}[Sing X], \Xi \mathbb{Z}[n] ) \,.

Using next the Dold-Kan adjunction this is

H 0Ch(Ch (X),[n]), \cdots \simeq H_0 Ch( Ch_\bullet(X), \mathbb{Z}[n] ) \,,

where

Ch (X):=N ((SingX)) Ch_\bullet(X) := N^\bullet(\mathbb{Z}(Sing X))

is the Moore complex of normalized chains of [SingX]\mathbb{Z}[Sing X]: this is the complex of singular chains, formal sums over \mathbb{Z} of simplices in XX.

This way singular cohomology is the abelian dual of singular homology.

Comparison to sheaf cohomology

If the topological space XX is semi-locally contractible (meaning: any open subset UXU\subset X has an open cover WW by open subsets W iUW_i\subset U that are contractible in UU), then the sheaf cohomology of XX is isomorphic to the singular cohomology of XX for any abelian group of coefficients.

This was proved in (Sella 16).

Discussion

A previous version of this entry led to the following discussion, which later led to extensive discussion by email. Partly as a result of this and similar discussions, there is now more information on how Kan complexes are \infty-groupoids at

References

Early references on (co)homology

The original references on chain homology/cochain cohomology and ordinary cohomology in the form of cellular cohomology:

  • Andrei Kolmogoroff, Über die Dualität im Aufbau der kombinatorischen Topologie, Recueil Mathématique 1(43) (1936), 97–102. (mathnet)

A footnote on the first page reads as follows, giving attribution to Alexander 35a, 35b:

Die Resultate dieser Arbeit wurden für den Fall gewöhnlicher Komplexe vom Verfasser im Frühling und im Sommer 1934 erhalten und teilweise an der Internationalen Konferenz für Tensoranalysis (Moskau) im Mai 1934 vorgetragen. Die hier dargestellte allgemeinere Theorie bildete den Gegenstand eines Vortrages, den der Verfasser an der Internationalen Topologischen Konferenz (Moskau, September 1935) hielt; bei letzterer Gelegenheit erfuhr er, dass ein grosser Teil dieser Resultate im Falle von Komplexen indessen von Herrn Alexander erhalten worden ist. Vgl. die inzwischen erschienenen Noten von Herrn Alexander in den «Proceedings of the National Academy of Sciences U.S.A.», 21, (1935), 509—512. Herr Alexander trug über seine Resultate ebenfalls an der Moskauer Topologischen Konferenz vor. Verallgemeinerungen für abgeschlossene Mengen und die Konstruktion eines Homologieringes für Komplexe und abgeschlossene Mengen, über welche der Verfasser ebenso an der Tensorkonferenz 1934 vorgetragen hat, werden in einer weiteren Publikation dargestellt. Diese weitere Begriffsbildungen sind übrigens ebenfalls von Herrn Alexander gefunden und teilweise in den erwähnten Noten publiziert.

The term “cohomology” was introduced by Hassler Whitney in

See also

The notion of singular cohomology is due to

The general abstract perspective on cohomology (subsuming sheaf cohomology, hypercohomology, non-abelian cohomology and indications of Whitehead-generalized cohomology) was essentially established in:

but probably known in one form or other before that.

General

Relation to sheaf cohomology:

A simplified proof using hypercovers can be found in

Last revised on February 21, 2021 at 02:27:25. See the history of this page for a list of all contributions to it.