V1-periodic Homotopy Groups of SO(n)

Author: Martin Bendersky,Donald M. Davis

Publisher: American Mathematical Soc.

ISBN: 0821835890

Category: Mathematics

Page: 90

View: 8275

DOWNLOAD NOW »

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.
Release

V1-periodic Homotopy Groups of SO(n)

Author: Martin Bendersky,Donald M. Davis

Publisher: American Mathematical Soc.

ISBN: 0821835890

Category: Mathematics

Page: 90

View: 7240

DOWNLOAD NOW »

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.
Release

Topology and Representation Theory

Conference on the Connections Between Topology and Representation Theory, May 1-5, 1992, Northwestern University

Author: Eric M. Friedlander,Mark E. Mahowald,American Mathematical Society

Publisher: American Mathematical Soc.

ISBN: 0821851659

Category: Mathematics

Page: 318

View: 6889

DOWNLOAD NOW »

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.
Release

Arrugas en el tiempo

Author: George Smoot,Keay Davidson

Publisher: Grano de Sal

ISBN: 6079824922

Category: Science

Page: 328

View: 9944

DOWNLOAD NOW »

Todo lo que somos y todo lo que nos rodea proviene de un mismo lugar y un mismo momento: el Big Bang. La cosmología contemporánea, esa disciplina en que la astronomía convive con la física cuántica y la relatividad general para estudiar el origen y la evolución temprana del universo, explica el surgimiento y la distribución de los cuerpos celestes y los elementos químicos. George Smoot y Keay Davidson presentan en este libro un recuento de los hitos que a lo largo del siglo XX transformaron nuestro modo de comprender el cosmos; es además una emocionante bitácora de las aportaciones del propio Smoot —con globos que ascienden a la estratosfera, aviones bombarderos adaptados para la exploración científica, severos viajes a la Antártida, todo ello aderezado con las rivalidades entre distintos grupos de investigación— para escudriñar en el fondo cósmico de microondas, como nunca se había hecho antes, en busca de pequeñas irregularidades —las "arrugas en el tiempo" del título— en la estructura del espacio-tiempo en los primeros momentos del Big Bang. Tal vez la contagiosa pasión que irradia este libro provenga de la certeza de George Smoot de que esos hallazgos fueron "como mirar a Dios" pues logró "vislumbrar el momento mismo de la creación". Por eso Stephen Hawking consideró que éste fue "el descubrimiento científico del siglo, si no es que de todos los tiempos".
Release