On the spectrum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.gif" display="inline" overflow="scroll"><mml:mi>b</mml:mi><mml:mstyle mathvariant="normal"><mml:mi>o</mml:mi></mml:mstyle><mml:mo>∧</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>tmf</mml:mi></mml:mstyle></mml:math>

The 2-primary Hopf invariant 1 elements in the stable homotopy groups of spheres form the most accessible family of elements. In this paper we explore some properties of the E∞ ring spectra obtained from certain iterated mapping cones by applying the free algebra functor. In fact, these are equivalent to Thom spectra over infinite loop spaces related to the classifying spaces BSO, BSpin, BString.We show that the homology of these Thom spectra are all extended comodule algebras of the form A * A(r) * P * over the dual Steenrod algebra A * with A * A(r) * F2 as an algebra retract.This suggests that these spectra might be wedges of module spectra over the ring spectra HZ, kO or tmf, however apart from the first case, we have no concrete results on this.Date: 16/04/2017 -version 8 arXiv:1503.05902 . 2010 Mathematics Subject Classification. Primary 55P43; Secondary 55P42, 57T05.

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

Powell

2015

Topology and its Applications

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A structure theorem for bounded-below modules over the subalgebra A (1) of the mod 2 Steenrod algebra generated by Sq 1 , Sq 2 is proved; this is applied to prove a classification theorem for a family of indecomposable A (1)-modules. The action of the A (1)-Picard group on this family is described, as is the behaviour of duality.The cohomology of dual Brown-Gitler spectra is identified within this family and the relation with members of the A (1)-Picard group is made explicit. Similarly, the cohomology of truncated projective spaces is considered within this classification; this leads to a conceptual understanding of various results within the literature. In particular, a unified approach to Ext-groups relevant to Adams spectral sequence calculations is obtained, englobing earlier results of Davis (for truncated projective spaces) and recent work of Pearson (for Brown-Gitler spectrum).2000 Mathematics Subject Classification. 19L41; 55S10. Key words and phrases. Steenrod algebra, indecomposable module, truncated projective space, Brown-Gitler spectra.The author is very grateful to an anonymous referee for remarks which have lead to an improvement in the presentation of the paper.

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On the spectrum bo∧tmf

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Connectedness of graphs arising from the dual Steenrod algebra

Connectedness of graphs arising from the dual Steenrod algebra

$$\mathcal {E}_\infty $$ E ∞ ring spectra and elements of Hopf invariant 1

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

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