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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.

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

Revised on February 24, 2016 13:02:04 by Urs Schreiber (193.136.135.97)