Homology rings of homotopy associative H-spaces

Abstract

MSC:Let X be a homotopy associative mod p H-space for p an odd prime. The homology H * (X; F p ) is an associative ring, but not necessarily commutative. We study conditions when [x, y] = 0 for x, y elements of H * (X; F p ). Under certain conditions [x, y] = 0 imply ad l (x, y) = 0 for l = p − 2 or p − 1. These methods can be used to prove results about homology commutators that were previously obtained using the adjoint action [H. Hamanaka, S. Hara, A. Kono, Adjoint action of Lie groups on the loop spaces and cohomology of exceptional Lie groups, Transform. Group Theory (1996) 44-50, Korea Adv. Inst. Sci. Tech.; A. Kono, K. Kozima, The adjoint action of a Lie group on the space of loops, J. Math. Soc. Japan 45 (3) (1993) 495-509; A. Kono, J. Lin, O. Nishimura, Characterization of the mod 3 cohomology of E 7 , Proc. Amer. Math. Soc. 131 (10) (2003) 3289-3295]. We also generalize results of Kane [R. Kane, Torsion in homotopy associative H-spaces, Illinois J. Math. 20 (1976) 476-485] to nonfinite mod p homotopy associative H-spaces.

The affirmative answer to Singer's conjecture on the algebraic transfer of rank four

“…We first consider item (ii) with , i.e., degree for all Let us recall that due to Sum [27], the dimension of is given by the following table: A monomial basis of is also given in the same paper [27]. Taking this basis, together with a computational technique similar to that of our works in [21, 22], we obtain that the -invariant space is trivial if and is 1-dimensional if As it is known, the dimensions of the domain of in degrees are determined by On the other hand, from the result by Lin [12], it follows that So, by the equalities and , we deduce that is bad for and that conjecture 1.5 holds for the degrees for …”

“…We need to compute the dimension of the -invariant . By Sum [27, 28], the dimension of the kenel of is determined as follows: Using this result and a similar computation as in [20, 22], we claim that is trivial if and has dimension if From these data, the inequality implies that On the other side, due to Lin [12], we find that It is known, the non-zero elements , for and for are detected by the fourth transfer. In fact, this could also be directly proved as our previous works [20–22] by using -level of .…”