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The mod 2 homology of infinite loopspaces

Nicholas J Kuhn and Jason B McCarty

Algebraic & Geometric Topology 13 (2013) 687–745

Applying mod 2 homology to the Goodwillie tower of the functor sending a spectrum X to the suspension spectrum of its 0th space, leads to a spectral sequence for computing H(ΩX; 2), which converges strongly when X is 0–connected. The E1 term is the homology of the extended powers of X, and thus is a well known functor of H(X; 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 E1.

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 has Dyer–Lashof operations induced from those on E1.

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; 2) for many spectra X, including suspension spectra and almost all Eilenberg–Mac Lane spectra. The E term of the algebraic spectral sequence has form and structure similar to E1, 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.

infinite loopspaces, Dyer–Lashof operations
Mathematical Subject Classification 2010
Primary: 55P47
Secondary: 55S10, 55S12, 55T99
Received: 14 February 2012
Revised: 1 October 2012
Accepted: 30 October 2012
Published: 24 March 2013
Nicholas J Kuhn
Department of Mathematics
University of Virginia
Kerchof Hall PO Box 400137
Charlottesville VA 22904-4137
Jason B McCarty
Department of Mathematics
Indiana University of Pennsylvania
Indiana PA 15705