Quadratic homology

Baues, Hans-Joachim

doi:10.1090/S0002-9947-99-02335-1

Quadratic homology
HTML articles powered by AMS MathViewer

by Hans-Joachim Baues PDF

Trans. Amer. Math. Soc. 351 (1999), 429-475 Request permission

Abstract:

We describe axioms for a ‘quadratic homology theory’ which generalize the classical axioms of homology. As examples we consider quadratic homology theories induced by 2-excisive homotopy functors in the sense of Goodwillie and the homology of a space with coefficients in a square group which generalizes the homology of a space with coefficients in an abelian group.

References

Arone, G. and Mahowald, M., The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Preprint (1995), Northwestern University.
Hans J. Baues, Obstruction theory on homotopy classification of maps, Lecture Notes in Mathematics, Vol. 628, Springer-Verlag, Berlin-New York, 1977. MR 0467748
Hans Joachim Baues, Algebraic homotopy, Cambridge Studies in Advanced Mathematics, vol. 15, Cambridge University Press, Cambridge, 1989. MR 985099, DOI 10.1017/CBO9780511662522
Hans Joachim Baues, Metastable homotopy and homology, Quaestiones Math. 14 (1991), no. 2, 161–178. MR 1107678
Hans Joachim Baues, Quadratic functors and metastable homotopy, J. Pure Appl. Algebra 91 (1994), no. 1-3, 49–107. MR 1255923, DOI 10.1016/0022-4049(94)90135-X
Hans Joachim Baues, Combinatorial homotopy and $4$-dimensional complexes, De Gruyter Expositions in Mathematics, vol. 2, Walter de Gruyter & Co., Berlin, 1991. With a preface by Ronald Brown. MR 1096295, DOI 10.1515/9783110854480
Hans-Joachim Baues, Homotopy type and homology, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1404516
Baues, H.J. and Conduché, D., On the 2-type of an iterated loop space, Preprint Max-Planck-Institut für Mathematik 95-112.
Baues, H.J. and Hartl, M. and Pirashvili, T., Quadratic categories and square rings, J. Pure Appl. Algebra 122, 1997, pp. 1–40.
Baues, H.J. and Pirashvili, T., Quadratic endofunctors of the catgeory of groups, Preprint MPI für Math. 95-55.
Baues, H.J. and Pirashvili, T., The universal coefficient theorem for quadratic functors, Preprint MPI für Math. 95-57.
A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573
F. R. Cohen, J. P. May, and L. R. Taylor, Splitting of certain spaces $CX$, Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 3, 465–496. MR 503007, DOI 10.1017/S0305004100055298
Daniel Conduché, Modules croisés généralisés de longueur $2$, Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983), 1984, pp. 155–178 (French, with English summary). MR 772056, DOI 10.1016/0022-4049(84)90034-3
Edward B. Curtis, Simplicial homotopy theory, Advances in Math. 6 (1971), 107–209 (1971). MR 279808, DOI 10.1016/0001-8708(71)90015-6
Albrecht Dold and Dieter Puppe, Homologie nicht-additiver Funktoren. Anwendungen, Ann. Inst. Fourier (Grenoble) 11 (1961), 201–312 (German, with French summary). MR 150183
Thomas G. Goodwillie, Calculus. I. The first derivative of pseudoisotopy theory, $K$-Theory 4 (1990), no. 1, 1–27. MR 1076523, DOI 10.1007/BF00534191
Thomas G. Goodwillie, Calculus. II. Analytic functors, $K$-Theory 5 (1991/92), no. 4, 295–332. MR 1162445, DOI 10.1007/BF00535644
Goodwillie, T.G., Calculus III: the Taylor series of a homotopy functor, in preparation.
Goodwillie, T.G., The calculus of functors, Preprint 1986 Brown University.
Goodwillie, T.G., Calculus I. The first derivative of a homotopy functor, Preprint Brown University.
Brayton Gray, Homotopy theory, Pure and Applied Mathematics, Vol. 64, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. An introduction to algebraic topology. MR 0402714
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Brenda Johnson, The derivatives of homotopy theory, Trans. Amer. Math. Soc. 347 (1995), no. 4, 1295–1321. MR 1297532, DOI 10.1090/S0002-9947-1995-1297532-6
Daniel M. Kan, A relation between $\textrm {CW}$-complexes and free c.s.s. groups, Amer. J. Math. 81 (1959), 512–528. MR 111036, DOI 10.2307/2372755
Daniel M. Kan, A combinatorial definition of homotopy groups, Ann. of Math. (2) 67 (1958), 282–312. MR 111032, DOI 10.2307/1970006
Saunders Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Band 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156879
George W. Whitehead, Recent advances in homotopy theory, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 5, American Mathematical Society, Providence, R.I., 1970. MR 0309097
George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508