# Stable Homotopy Theory Book

Part of the Lecture Notes in Mathematics book series (LNM, volume 3)

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.Department of MathematicsUniversity of ManchesterUK

### Bibliographic information

• Book Title Stable Homotopy Theory
• Series Title Lecture Notes in Mathematics
• 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
• Softcover ISBN 978-3-662-15944-6
• eBook ISBN 978-3-662-15942-2
• Series ISSN 0075-8434
• Series E-ISSN 1617-9692
• Edition Number 1
• Number of Pages III, 77
• Number of Illustrations 3 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site