The BP homology of 𝐻-spaces

Kane, Richard

doi:10.1090/S0002-9947-1978-0478143-9

The BP homology of $H$-spaces
HTML articles powered by AMS MathViewer

by Richard Kane PDF

Trans. Amer. Math. Soc. 241 (1978), 99-119 Request permission

Abstract:

The BP homology of 1-connected H-spaces whose loop space is torsion free is studied. It is shown that localizing in a suitable manner kills all torsion in $B{P_{\ast }}(X)$. Similar results are then obtained for the bordism of X. Finally the K-theory of X is shown to have no p torsion.

References

J. F. Adams, Lectures on generalised cohomology, Category Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three), Springer, Berlin, 1969, pp. 1–138. MR 0251716

Stable homotopy and generalized homology

Raoul Bott, The space of loops on a Lie group, Michigan Math. J. 5 (1958), 35–61. MR 102803
Edgar H. Brown Jr. and Franklin P. Peterson, A spectrum whose $Z_{p}$ cohomology is the algebra of reduced $p^{th}$ powers, Topology 5 (1966), 149–154. MR 192494, DOI 10.1016/0040-9383(66)90015-2
William Browder, Torsion in $H$-spaces, Ann. of Math. (2) 74 (1961), 24–51. MR 124891, DOI 10.2307/1970305
William Browder, On differential Hopf algebras, Trans. Amer. Math. Soc. 107 (1963), 153–176. MR 145530, DOI 10.1090/S0002-9947-1963-0145530-7
William Browder, Higher torsion in $H$-spaces, Trans. Amer. Math. Soc. 108 (1963), 353–375. MR 155326, DOI 10.1090/S0002-9947-1963-0155326-8
Allan Clark, Homotopy commutativity and the Moore spectral sequence, Pacific J. Math. 15 (1965), 65–74. MR 177416, DOI 10.2140/pjm.1965.15.65
Allan Clark, Hopf algebras over Dedekind domains and torsion in $H$-spaces, Pacific J. Math. 15 (1965), 419–426. MR 189026, DOI 10.2140/pjm.1965.15.419
Francis Clarke, On the $K$-theory of the loop space of a Lie group, Proc. Cambridge Philos. Soc. 76 (1974), 1–20. MR 348745, DOI 10.1017/s0305004100048672
P. E. Conner and E. E. Floyd, The relation of cobordism to $K$-theories, Lecture Notes in Mathematics, No. 28, Springer-Verlag, Berlin-New York, 1966. MR 0216511, DOI 10.1007/BFb0071091

On general cohomology

Michiel Hazewinkel, A universal formal group and complex cobordism, Bull. Amer. Math. Soc. 81 (1975), no. 5, 930–933. MR 371909, DOI 10.1090/S0002-9904-1975-13891-2

On the K-theory of Lie groups

6

David Copeland Johnson and W. Stephen Wilson, Projective dimension and Brown-Peterson homology, Topology 12 (1973), 327–353. MR 334257, DOI 10.1016/0040-9383(73)90027-X
Richard Kane, On loop spaces without $p$ torsion, Pacific J. Math. 60 (1975), no. 1, 189–201. MR 383396, DOI 10.2140/pjm.1975.60.189
Richard Kane, On loop spaces without $p$ torsion, Pacific J. Math. 60 (1975), no. 1, 189–201. MR 383396, DOI 10.2140/pjm.1975.60.189
Richard Kane, The module of indecomposables for finite $H$-spaces, Trans. Amer. Math. Soc. 222 (1976), 303–318. MR 423341, DOI 10.1090/S0002-9947-1976-0423341-1

Rational BP operations and the chern character

James P. Lin, $H$-spaces with finitely generated cohomology algebras, Bull. Amer. Math. Soc. 80 (1974), 1233–1238. MR 367993, DOI 10.1090/S0002-9904-1974-13694-3

Torsion in H-spaces

John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171. MR 99653, DOI 10.2307/1969932
John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MR 174052, DOI 10.2307/1970615
Ted Petrie, The weakly complex bordism of Lie groups, Ann. of Math. (2) 88 (1968), 371–402. MR 267602, DOI 10.2307/1970719
Daniel Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969), 1293–1298. MR 253350, DOI 10.1090/S0002-9904-1969-12401-8
Raphael Zahler, The Adams-Novikov spectral sequence for the spheres, Ann. of Math. (2) 96 (1972), 480–504. MR 319197, DOI 10.2307/1970821