nLab
singular simplicial complex

Contents

Idea

This construction ‘probes’ a space XX by mapping geometric simplices into it. It is one of the classical approaches to determining invariants of the homotopy type of the space.

Definition

The singular simplicial complex S (X)S_\bullet(X) of a topological space XX is the nerve of XX with respect to the standard cosimplicial topological space Δ Top:ΔTop\Delta_{Top} : \Delta \to Top. It is thus the simplicial set, S (X)S_\bullet(X), having

S n(X)=Hom Top(Δ Top n,X). S_n(X) = Hom_{Top}(\Delta_{Top}^n, X) \,.

as its set of nn-simplices, and fairly obvious faces and degeneracy mappings obtains by restriction along the structural maps of Δ Top:ΔTop\Delta_{Top} : \Delta \to Top. This is always a Kan complex and as such has the interpretation of the fundamental ∞-groupoid Π(X)\Pi(X) of XX.

The nn-simplices of this are just singular n-simplices generalising paths in XX. (The term -singular_ is used because there is no restriction that the continuous function used should be an embedding, as would be the case in, for instance, a triangulation where a simplex in the underlying simplicial complex corresponds to an embedding of a simplex.)

Properties

Relation to geometric realization

Together with its adjointgeometric realization :sSetTop|-| : sSet \to Top – the functor Sing:TopsSetSing : Top \to sSet is part of the Quillen equivalence between the model structure on topological spaces and the model structure on simplicial sets that is sometimes called the homotopy hypothesis-theorem.

Relation to ordinary (co)homology

Forming from the singular simplicial complex Sing(X)Sing(X) first the free simplicial abelian group [Sing(X)]\mathbb{Z}[Sing(X)] and then under the Dold-Kan correspondence the correspondin normalized chain complex yields this chain complex which computes the singular homology of XX.

Revised on November 15, 2013 00:32:13 by Urs Schreiber (82.169.114.243)