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singular cohomology

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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.

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

Revised on October 29, 2013 09:16:50 by Adeel Khan (132.252.63.205)