The mod 2 homology of infinite loopspaces

Kuhn, Nicholas; McCarty, Jason

doi:10.2140/agt.2013.13.687

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1472-2739 (online)

ISSN 1472-2747 (print)

Author Index

To Appear

Other MSP Journals

The mod 2 homology of infinite loopspaces

Nicholas J Kuhn and Jason B McCarty

Algebraic & Geometric Topology 13 (2013) 687–745

DOI: 10.2140/agt.2013.13.687

Abstract

Applying mod 2 homology to the Goodwillie tower of the functor sending a spectrum $X$ to the suspension spectrum of its $0^{t h}$ space, leads to a spectral sequence for computing $H_{*} (Ω^{\infty} X; ℤ ∕ 2)$ , which converges strongly when $X$ is 0–connected. The $E^{1}$ term is the homology of the extended powers of $X$ , and thus is a well known functor of $H_{*} (X; ℤ ∕ 2)$ , including structure as a bigraded Hopf algebra, a right module over the mod 2 Steenrod algebra $A$ , and a left module over the Dyer–Lashof operations. This paper is an investigation of how this structure is transformed through the spectral sequence.

Hopf algebra considerations show that all pages of the spectral sequence are primitively generated, with primitives equal to a subquotient of the primitives in $E^{1}$ .

We use an operad action on the tower, and the Tate construction, to determine how Dyer–Lashof operations act on the spectral sequence. In particular, $E^{\infty}$ has Dyer–Lashof operations induced from those on $E^{1}$ .

We use our spectral sequence Dyer–Lashof operations to determine differentials that hold for any spectrum $X$ . The formulae for these universal differentials then lead us to construct an algebraic spectral sequence depending functorially on an $A$ –module $M$ . The topological spectral sequence for $X$ agrees with the algebraic spectral sequence for $H_{*} (X; ℤ ∕ 2)$ for many spectra $X$ , including suspension spectra and almost all Eilenberg–Mac Lane spectra. The $E^{\infty}$ term of the algebraic spectral sequence has form and structure similar to $E^{1}$ , but now the right $A$ –module structure is unstable. Our explicit formula involves the derived functors of destabilization as studied in the 1980’s by W Singer, J Lannes and S Zarati, and P Goerss.

Keywords

infinite loopspaces, Dyer–Lashof operations

Mathematical Subject Classification 2010

Primary: 55P47

Secondary: 55S10, 55S12, 55T99

References

Publication

Received: 14 February 2012

Revised: 1 October 2012

Accepted: 30 October 2012

Published: 24 March 2013

Authors

Nicholas J Kuhn
Department of Mathematics University of Virginia Kerchof Hall PO Box 400137 Charlottesville VA 22904-4137 USA
http://pi.math.virginia.edu/Faculty/Kuhn
Jason B McCarty
Department of Mathematics Indiana University of Pennsylvania Indiana PA 15705 USA

Volume 13, issue 2 (2013)