An integral version of the Brown-Gitler spectrum

Abstract

Abstract. In this paper, certain spectra B^k) are studied whose behavior qualifies them as being integral versions of the Brown-Gitler spectra B(k). The bulk of our work emphasizes the similarities between B^(k) and B(k), shown mainly using the techniques of Brown and Gitler. In analyzing the homotopy type of B^k), we provide a free resolution over the Steenrod algebra for its cohomology and study its Adams spectral sequence. We also list conditions which characterize it at the prime 2. The paper begins, however, on a somewhat different topic, namely, the construction of a configuration space model for Q2(S3(3)) and other related spaces.Introduction. In a paper published in 1973, E. H. Brown, Jr., and S. Gitler described a procedure by which one might conceivably find new characteristic classes for smooth rt-dimensional manifolds [3]. Their approach was based on examining the way that certain cohomology operations act on the stable normal bundle. Much of the motivation for their work stemmed from the problem of immersing manifolds in Euclidean space. For example, suppose that M is an n-manifold and let U denote the mod 2 Thorn class of its stable normal bundle. From standard properties of the Steenrod squares, it follows that if Sq' U ¥= 0, then M cannot be immersed in Ra+i~1.Brown and Gitler's idea was to take all those elements of the mod 2 Steenrod algebra which universally vanish on the Thorn classes of ^-manifolds and use these "primary" operations as the foundation for a coherent system of higher order cohomology operations which might then prove useful, say, to detect nonimmersions.

Integral Brown-Gitler Spectra

“…The first two parts of the following theorem, which states the basic properties of integral Brown-Gitler spectra, now follow immediately from Theorem 1.3. Part (iii) is not easily proved from our perspective, but does follow from a straightforward adaptation of the proof of [Sh,5.1], which was given only for p = 2. This adaptation requires a map from Bi(n) into the mod p BrownGitler spectrum B(pn + 1) inducing the obvious inclusion in homology.…”

Truncated projective spaces, Brown–Gitler spectra and indecomposable A(1)-modules

“…With topological applications in mind and to fix conventions, recall the families B(k), B 0 (k) indexed by N, where B(k) is the kth Brown-Gitler spectrum [BG73,GLM93] and B 0 (k) is used to denote the kth integral Brown-Gitler spectrum (cf. [Shi84], where a different indexing is used); the notation B 0 follows [Pea14]. The indexing below follows that of [GLM93], in particular there are homotopy equivalences B(2k) ≃ B(2k + 1) and B 0 (2k) ≃ B 0 (2k + 1), so consideration is limited to the even-indexed spectra.…”

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…”