1. Front Matter
Pages N2-iii
Pages 1-3
Pages 4-21
Pages 22-37
Pages 38-57
Pages 58-68
Pages 69-73
8. Back Matter
Pages 74-77

### Introduction

Before I get down to the business of exposition, I'd like to offer a little motivation. I want to show that there are one or two places in homotopy theory where we strongly suspect that there is something systematic going on, but where we are not yet sure what the system is. The first question concerns the stable J-homomorphism. I recall that this is a homomorphism J: ~ (SQ) ~ ~S = ~ + (Sn), n large. r r r n It is of interest to the differential topologists. Since Bott, we know that ~ (SO) is periodic with period 8: r 6 8 r = 1 2 3 4 5 7 9· . · Z o o o z On the other hand, ~S is not known, but we can nevertheless r ask about the behavior of J. The differential topologists prove: 2 Th~~: If I' = ~ - 1, so that 'IT"r(SO) ~ 2, then J('IT"r(SO)) = 2m where m is a multiple of the denominator of ~/4k th (l\. being in the Pc Bepnoulli numher.) Conject~~: The above result is best possible, i.e. J('IT"r(SO)) = 2m where m 1s exactly this denominator. status of conJectuI'e ~ No proof in sight. Q9njecture Eo If I' = 8k or 8k + 1, so that 'IT"r(SO) = Z2' then J('IT"r(SO)) = 2 , 2 status of conjecture: Probably provable, but this is work in progl'ess.

### Keywords

Division Homological algebra Homotopie Homotopy Morphism behavior homomorphism homotopy theory proof system

#### Authors and affiliations

• 1
1. 1.Department of MathematicsUniversity of ManchesterUK

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-662-15942-2
• Copyright Information Springer-Verlag Berlin Heidelberg 1964
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-662-15944-6
• Online ISBN 978-3-662-15942-2
• Series Print ISSN 0075-8434
• Series Online ISSN 1617-9692