V1 periodic Homotopy Groups of SO n

V1 periodic Homotopy Groups of SO n

Library of Congress Cataloging-in-Publication Data Bendersky, Martin, 1945– v1-
periodic homotopy groups of SO(n) / Martin Bendersky, Donald M. Davis. p. cm. –
(Memoirs of the American Mathematical Society, ISSN 0065-9266; no.

Author: Martin Bendersky

Publisher: American Mathematical Soc.

ISBN: 9780821835890

Category: Mathematics

Page: 90

View: 103

We compute the 2-primary $v_1$-periodic homotopy groups of the special orthogonal groups $SO(n)$. The method is to calculate the Bendersky-Thompson spectral sequence, a $K_*$-based unstable homotopy spectral sequence, of $\operatorname{Spin}(n)$. The $E_2$-term is an Ext group in a category of Adams modules. Most of the differentials in the spectral sequence are determined by naturality from those in the spheres. The resulting groups consist of two main parts. One is summands whose order depends on the minimal exponent of 2 in several sums of binomial coefficients times powers. The other is a sum of roughly $[\log_2(2n/3)]$ copies of ${\bold Z}/2$. As the spectral sequence converges to the $v_1$-periodic homotopy groups of the $K$-completion of a space, one important part of the proof is that the natural map from $\operatorname{Spin}(n)$ to its $K$-completion induces an isomorphism in $v_1$-periodic homotopy groups.
Categories: Mathematics

V1 periodic Homotopy Groups of SO n

V1 periodic Homotopy Groups of SO n

Author: Martin Bendersky

Publisher: American Mathematical Soc.

ISBN: 9780821835890

Category: Mathematics

Page: 90

View: 705

We compute the 2-primary $v_1$-periodic homotopy groups of the special orthogonal groups $SO(n)$. The method is to calculate the Bendersky-Thompson spectral sequence, a $K_*$-based unstable homotopy spectral sequence, of $\operatorname{Spin}(n)$. The $E_2$-term is an Ext group in a category of Adams modules. Most of the differentials in the spectral sequence are determined by naturality from those in the spheres. The resulting groups consist of two main parts. One is summands whose order depends on the minimal exponent of 2 in several sums of binomial coefficients times powers. The other is a sum of roughly $[\log_2(2n/3)]$ copies of ${\bold Z}/2$. As the spectral sequence converges to the $v_1$-periodic homotopy groups of the $K$-completion of a space, one important part of the proof is that the natural map from $\operatorname{Spin}(n)$ to its $K$-completion induces an isomorphism in $v_1$-periodic homotopy groups.
Categories: Mathematics

Topology and Representation Theory

Topology and Representation Theory

Math. Z, 182:553–568, 1983. [Co90] F.R. Cohen. A note concerning the v1-
periodic homotopy of odd spheres. ... J. Math. Kyoto Univ., 31:43–70, 1991. [Ko03
] S.O. Kochman. Homology of the classical groups over the Dyer-Lashof algebra.
Trans. ... Memoirs of the American Mathematical Society No. 232. ... [Ya80] N.
Yagita.

Author: Eric M. Friedlander

Publisher: American Mathematical Soc.

ISBN: 9780821851654

Category: Mathematics

Page: 318

View: 252

During 1991-1992, Northwestern University conducted a special emphasis year on the topic, 'The connections between topology and representation theory'. Activities over the year culminated in a conference in May 1992 which attracted over 120 participants. Most of the plenary lectures at the conference were expository and designed to introduce current trends to graduate students and nonspecialists familiar with algebraic topology. This volume contains refereed papers presented or solicited at the conference; one paper is based on a seminar given during the emphasis year.
Categories: Mathematics

Lehrbuch der Topologie

Lehrbuch der Topologie

Author: Herbert Seifert

Publisher:

ISBN: OCLC:493389807

Category: Topology

Page: 353

View: 425

Categories: Topology