A chain functor for bordism

Kochman, Stanley O.

doi:10.1090/S0002-9947-1978-0488031-X

A chain functor for bordism
HTML articles powered by AMS MathViewer

by Stanley O. Kochman

Trans. Amer. Math. Soc. 239 (1978), 167-196

DOI: https://doi.org/10.1090/S0002-9947-1978-0488031-X

PDF | Request permission

Abstract:

Chains of differential graded abelian monoids are defined for bordism and cobordism theories. These chains are used to define matric Massey products and can be filtered so as to define the Adams spectral sequence. From this point of view, we prove three basic theorems which show how Massey products behave in the Adams spectral sequence.

References

J. F. Adams, On the structure and applications of the Steenrod algebra, Comment. Math. Helv. 32 (1958), 180–214. MR 96219, DOI 10.1007/BF02564578
J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1974. MR 0402720
J. C. Alexander, Cobordism Massey products, Trans. Amer. Math. Soc. 166 (1972), 197–214. MR 293623, DOI 10.1090/S0002-9947-1972-0293623-1
D. W. Anderson, Chain functors and homology theories, Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle, Wash., 1971) Lecture Notes in Math., Vol. 249, Springer, Berlin, 1971, pp. 1–12. MR 0339132
M. F. Atiyah, Bordism and cobordism, Proc. Cambridge Philos. Soc. 57 (1961), 200–208. MR 126856, DOI 10.1017/s0305004100035064
R. O. Burdick, P. E. Conner, and E. E. Floyd, Chain theories and their derived homology, Proc. Amer. Math. Soc. 19 (1968), 1115–1118. MR 233346, DOI 10.1090/S0002-9939-1968-0233346-2
Jean Cerf, Topologie de certains espaces de plongements, Bull. Soc. Math. France 89 (1961), 227–380 (French). MR 140120, DOI 10.24033/bsmf.1567
Joel M. Cohen, Stable homotopy, Lecture Notes in Mathematics, Vol. 165, Springer-Verlag, Berlin-New York, 1970. MR 0273608, DOI 10.1007/BFb0059783
Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, N.J., 1952. MR 0050886, DOI 10.1515/9781400877492

The symplectic cobordism ring

Anders Kock, Leif Kristensen, and Ib Madsen, Cochain functors for general cohomology theories. I, II, Math. Scand. 20 (1967), 131–150; 151–176. MR 214056, DOI 10.7146/math.scand.a-10827
R. Lashof, Poincaré duality and cobordism, Trans. Amer. Math. Soc. 109 (1963), 257–277. MR 156357, DOI 10.1090/S0002-9947-1963-0156357-4
Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215
J. Peter May, Matric Massey products, J. Algebra 12 (1969), 533–568. MR 238929, DOI 10.1016/0021-8693(69)90027-1

ring spaces and

ring spectra

John Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, N.J., 1965. Notes by L. Siebenmann and J. Sondow. MR 0190942, DOI 10.1515/9781400878055
Gerald J. Porter, Higher products, Trans. Amer. Math. Soc. 148 (1970), 315–345. MR 256397, DOI 10.1090/S0002-9947-1970-0256397-4
James Dillon Stasheff, Homotopy associativity of $H$-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 293–312. 108 (1963), 275-292; ibid. MR 0158400, DOI 10.1090/S0002-9947-1963-0158400-5
Robert E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. Mathematical notes. MR 0248858