Ordinary homology is generalized homology with respect to an Eilenberg-MacLane spectrum , and often understood over .
Equivalently this is computed by singular homology with coefficients in .
Relation to homotopy groups
Ordinary homology with coefficients in of a topological space serves to approximate the homotopy groups of .
See for instance at Hurewicz theorem
Description in terms of higher linear algebra
We discuss (twisted) ordinary homology and ordinary cohomology in terms of sections of (∞,1)-module bundles over the Eilenberg-MacLane spectrum.
Let be a commutative ring. Write
for the Eilenberg-MacLane spectrum of , canonically regarded as an E-∞ ring. Write
for the (∞,1)-category of (∞,1)-modules over .
This is the statement of the stable Dold-Kan correspondence, see at (∞,1)-category of (∞,1)-modules – Properties – Stable Dold-Kan correspondence.
This is the main theorem in (Block-Smith 09).
for the (∞,1)-colimit functor.
for the flat (∞,1)-module bundle which is constant on the chain complex concentrated on in degree 0, the tensor unit in .
This is a classical basic (maybe folklore) statement. Here is one way to see it in full detail.
First notice that the (∞,1)-colimit of functors out of ∞-groupoids and constant on the tensor unit in is by definition the (∞,1)-tensoring operation of over ∞Grpd. Now if we find a presentation of by a simplicial model category the by the dicussion at (∞,1)-colomit – Tensoring and cotensoring – Models this (∞,1)-tensoring is given by the left derived functor of the sSet-tensoring in that simplicial model category.
To obtain this, use prop. 1 and then the discussion at model structure on chain complexes in the section Projective model structure on unbounded chain complexes which says that there is a simplicial model category structure on the category of simplicial objects in the category of unbounded chain complexes which models , and whose weak equivalences are those morphisms that produce quasi-isomorphism under the total chain complex functor.
In summary it follows that with any simplicial set representing (under the homotopy hypothesis-theorem) we have
where on the right we have the coend over the simplex category of the tensoring (of simplicial sets with simplicial objects in the category of unbounded chain complexes) of the standard cosimplicial simplex with the simplicial diagram constant on the tensor unit chain complex.
The result on the right is manifestly, by the very definition of singular homology, under the ordinary Dold-Kan correspondence the chain complex of singular simplices:
If is a Poincaré duality space of dimension , then is a dualizable object which is almost self-dual except for a degree twist of (itself) degree 0:
See also at Dold-Thom theorem.
Ordinary homology spectra split
see at ordinary homology spectra split
- Robert Switzer, chapter 10 of Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.
The Riemann-Hilbert correspondence/de Rham theorem for -modules is established in