Integral Brown-Gitler Spectra

Abstract

ABSTRACT. A Thorn spectrum model for integral Brown-Gitler spectra is established and shown to have a multiplicativeproperty. This clarifies certain aspects of an earlier application to splitting bo A bo. Statement of results.Brown-Gitler spectra have had many important applications in homotopy theory, most notably in [Ml and Cl]. They were originally constructed in [BG] by a complicated Postnikov argument, but a Thom spectrum model suggested in [Ml] and established to be correct in [C2] made them more down-to-earth.Integral Brown-Gitler spectra at the prime 2 were introduced in [M2], where they were useful in a splitting of bo A bo. A Thom spectrum model was suggested there, and an expanded account, including both Thom spectrum and Postnikov models, was presented in [Sh]. The odd-primary version of the Thom space model was discussed in [Ka]. In none of these is the base space for these Thom spectra explicitly defined. The purpose of this paper is to clarify the Thom spectrum model of integral Brown-Gitler spectra.Recall that there is an isomorphism of Hopf algebras The space Q2S3 admits an increasing filtration by spaces F"n253, due to May and Milgram [May, Mil], such that Ht,(Fn02S3) C Hm(Q2S3) is the span of monomials of weight < n [CLM, p. 239].Let S3 (3) denote the 3-connected cover of S3. Then there is a homotopy fibration n253(3)^n2s3^sx.n253(3) was called W in [DGM and M2]. Using the multiplication on Ü2S3, one easily deduces U2S3 ~ S1 x Q2S3(3), and so ¿í,(n253(3)) C ¿L,(fi2S3) is the span

On the E2‐term of the bo‐Adams spectral sequence

“…Since v 5 0 i 2 is above the 1/3 line, we deduce that v 0 h 2 3 is v 1 -periodic. Because it is located in the region of Ext where multiplication by v 8 1 is well defined, we deduce that in fact v 0 h 2 3 is v 8 1 -periodic.…”

“…The classes (40, 15 : 1) and (39, 16 : 3) are v 4 1 -periodic on the E 1+ -term of the AKSS. By Lemma 8.6, this differential is v 4 1 -periodic (and, therefore, v 8 1 -periodic). The classes (39, 14 : 1) and (38, 15 : 3) are v 8 1 -periodic on the E 1+ -term.…”

“…Let B i denote the ith integral Brown-Gitler spectrum (see, for example, [8]). The integral Brown-Gitler spectra are a sequence of finite complexes whose colimit is the integral Eilenberg-Maclane spectrum:…”

From elliptic curves to homotopy theory

“…where B(k) is the integral Brown Gitler spectrum [1] and bo k is the bo Brown Gitler spectrum. This sequence is constructed in [5].…”

On the spectrum bo∧tmf

“…Such spectra have been studied extensively in the literature (see [6][7][8], for example). In particular, Mahowald [4] demonstrated…”