nLab
generalized homology

Contents

Idea

Every spectrum KK is the coefficient object of a generalized cohomology theory and dually of a generalized homology theory.

For K=HRK = H R an Eilenberg-MacLane spectrum this reduces to ordinary homology

References

Original articles include

See also

  • Friedrich Bauer, Classifying spectra for generalized homology theories Annali di Maternatica pura ed applicata (IV), Vol. CLXIV (1993), pp. 365-399

  • Friedrich Bauer, Remarks on universal coefficient theorems for generalized homology theories Quaestiones Mathematicae Volume 9, Issue 1 & 4, 1986, Pages 29 - 54

A general construction of homologies by “geometric cycles” similar to the Baum-Douglas geometric cycles for K-homology is discussed in

  • S. Buoncristiano, C. P. Rourke and B. J. Sanderson, A geometric approach to homology theory, Cambridge Univ. Press, Cambridge, Mass. (1976)

Further generalization of this to bivariant cohomology theories is in

  • Martin Jakob, Bivariant theories for smooth manifolds, Applied Categorical Structures 10 no. 3 (2002)

Revised on February 23, 2014 03:15:57 by Urs Schreiber (77.80.20.34)