×

zbMATH — the first resource for mathematics

The projective plane of an \(H\)-pairing. (English) Zbl 0757.55011
Summary: The projective plane of an \(H\)-space has played an important role in the theory of \(H\)-spaces. In particular, if the cohomology of a given \(H\)- space is primitively generated, the cohomology of its projective plane has a simple structure. In this paper we discuss the projective plane of an \(H\)-pairing which is a generalized concept of an \(H\)-space. This extension gives us a method to construct a projective plain-like space out of an \(H\)-space. This space turns out to be useful even if the cohomology of the \(H\)-space is not primitively generated. As an application we prove some facts on the action of the squaring operations on the cohomology of a finite \(H\)-space.

MSC:
55P45 \(H\)-spaces and duals
55S10 Steenrod algebra
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adams, J.F., H-spaces with few cells, Topology, 1, 67-72, (1961) · Zbl 0109.41402
[2] Browder, W.; Thomas, E., On the projective plane of an H-space, Illinois J. math., 7, 492-502, (1963) · Zbl 0136.43905
[3] Iwase, N., H-spaces with generating subspaces, Proc. roy. soc. Edinburgh sect. A, 111, 199-211, (1989) · Zbl 0674.55007
[4] Kane, R.M., On loop spaces without p torsion, Pacific J. math., 60, 189-201, (1975) · Zbl 0322.55012
[5] Kane, R.M., Implications in Morava K-theory, Mem. amer. math. soc., 340, (1986) · Zbl 0585.57024
[6] Kane, R.M., The homology of Hopf spaces, () · Zbl 0577.55006
[7] Lin, J.P., Torsion in H-spaces, I, Ann. of math., 103, 457-487, (1976) · Zbl 0308.55008
[8] Lin, J.P., Torsion in H-spaces, II, Ann. of math., 107, 41-88, (1978) · Zbl 0388.55008
[9] Lin, J.P., Higher order operations in the mod 2 cohomology of finite H-spaces, Amer. J. math., 105, 855-937, (1983) · Zbl 0532.55012
[10] Lin, J.P., Steenrod connections and connectivity in H-spaces, Mem. amer. math. soc., 369, (1987) · Zbl 0625.55004
[11] Stasheff, J.D., On homotopy abelian H-spaces, Math. proc. Cambridge philos. soc., 57, 734-745, (1961) · Zbl 0106.16504
[12] Stasheff, J.D., Homotopy associativity of H-spaces, I, Trans. amer. math. soc., 108, 275-292, (1963) · Zbl 0114.39402
[13] Sugawara, M., On the homotopy-commutativity of groups and loop spaces, Mem. college sci. univ. of Kyoto ser. A, 33, 257-269, (1960) · Zbl 0113.16903
[14] Thomas, E., On the mod 2 cohomology of certain H-spaces, Comment. math. helv., 37, 132-140, (1963) · Zbl 0108.35705
[15] Thomas, E., Steenrod squares and H-spaces: II, Ann. of math., 81, 473-495, (1965) · Zbl 0127.13503
[16] Zabrodsky, A., Secondary cohomology operations in the module of indecomposables, (), 657-672 · Zbl 0224.55025
[17] Zabrodsky, A., Cohomology operations and homotopy commutative H-spaces, (), 308-317 · Zbl 0205.27603
[18] Zabrodsky, A., Secondary operations in the cohomology of H-spaces, Illinois J. math., 15, 648-655, (1971) · Zbl 0219.55009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.