Vol. 87, No. 2, 1980

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ISSN: 0030-8730
On the stable splitting of bo bo and torsion operations in connective K-theory

Gunnar Carlsson

Vol. 87 (1980), No. 2, 283–297
Abstract

The real connective K-theory spectrum, bo, has been shown to be a useful spectrum in homotopy theory. In particular, the bo-homology Adams Spectral Sequence, based on the cofiber sequence

                -                  -   -                      -   -  -
S0      ←−      bo       ← −       bo∧ bo         ←−          bo ∧ bo ∧bo
↘        ↗      ↘       -  ↗           ↘      -   -   ↗              ↘
bo              bo∧ bo                bo ∧bo ∧bo
(A)

has been used extensively by Mahowald in his work on the image of the J-homomorphism. One of the problems encountered with the bo-spectrum is that, unlike the mod 2 Eilenberg-Maclane spectrum, bobo does not split as a wedge of suspensions of bo itself. However, Mahowald and Milgram have obtained a splitting

bo∧ bo ≃ X ∨ G
(B)

where X is a wedge of spaces intimately related with bo itself, and G is a wedge of mod 2 Eilenberg-MacLane spectra. In this paper, we determine the structure of G, i.e., we calculate the number of Eilenberg-Maclane summands occuring in each dimension.

Mathematical Subject Classification 2000
Primary: 55N15
Secondary: 55T15
Milestones
Received: 23 October 1978
Published: 1 April 1980
Authors
Gunnar Carlsson