nLab stable homotopy homology theory

Idea

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

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

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

This is known as the Hurewicz homomorphism

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

This is known as the Boardman homomorphism.

Related concepts

• ordinary homology

• bordism homology theory

• stable cohomotopy theory

References

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