nLab
stable homotopy homology theory

Contents

Context

Stable Homotopy theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The generalized homology theory which is represented by the sphere spectrum 𝕊\mathbb{S} is usually called stable homotopy, since the homology groups that it assigns to a suitable topological space XX are just the stable homotopy group of XX:

H (X,𝕊)=𝕊 (X)π (𝕊X +)=π st(X). H_\bullet(X,\mathbb{S}) \;=\; \mathbb{S}_\bullet(X) \coloneqq \pi_\bullet\left( \mathbb{S} \wedge X_+ \right) \;=\; \simeq \pi_\bullet^{st}(X) \,.

Hence the coefficient cohomology ring of stable homotopy homology theory (its value on the point) is the stable homotopy groups of spheres. This highlights that stable homotopy homology of any space XX is extremely hard, or impossible, to completely analyze, since this is true already for the coefficient ring over the point.

Beware that the term stable homotopy theory, which would seem to be the canonical name for this generalized homology theory, traditionally refers instead to the general homotopy theory of spectra. The full term stable homotopy homology theory is used for emphasis, but clunky in practice.

The dual generalized cohomology theory, co-represented by the sphere spectrum, is called stable cohomotopy theory.

Properties

Relation to other homology theories

Along the canonical morphism of spectra 𝕊HR\mathbb{S} \to H R from the sphere spectrum to any Eilenberg-MacLane spectrum of a ring RR (which is the unit map of HRH R in the (infinity,1)-category of E-infinity ring spectra) stable homotopy homology maps to ordinary homology with coefficients in RR (given notably by singular homology).

This is known as the Hurewicz homomorphism

More generally, for EE an E-infinity ring spectrum, the unit 𝕊E\mathbb{S} \to E induces a natural transformation from stable homotopy homology theory to the generalized homology theory co-represented by EE.

This is known as the Boardman homomorphism.

References

  • Akhil Mathew, Torsion exponents in stable homotopy and the Hurewicz homomorphism, Algebr. Geom. Topol. 16 (2016) 1025-1041 (arXiv:1501.07561)

Created on September 22, 2018 at 05:40:58. See the history of this page for a list of all contributions to it.