Transfer, symmetric groups, and stable homotopy theory

1973 ◽  
pp. 244-255 ◽  
Author(s):  
Stewart B. Priddy
Keyword(s):  
Homotopy Theory ◽  
Symmetric Groups ◽  
Stable Homotopy ◽  
Stable Homotopy Theory

On the transfer in the homology of symmetric groups

1978 ◽  
Vol 83 (1) ◽  
pp. 91-101 ◽  
Author(s):  
Daniel S. Kahn ◽  
Stewart B. Priddy
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Homotopy Theory ◽  
Symmetric Groups ◽  
Characterization Theorem ◽  
Group Cohomology ◽  
Stable Homotopy ◽  
Stable Homotopy Theory ◽  
Transfer Homomorphism ◽  
Image Position

The transfer has long been a fundamental tool in the study of group cohomology. In recent years, symmetric groups and a geometric version of the transfer have begun to play an important role in stable homotopy theory (2, 5). Thus, motivated by geometric considerations, we have been led to investigate the transfer homomorphismin group homology, where n is the nth symmetric group, (n, p) is a p-Sylow sub-group and simple coefficients are taken in /p (the integers modulo a prime p). In this paper, we obtain an explicit characterization (Theorem 3·8) of this homomorphism. Roughly speaking, elements in H*(n) are expressible in terms of the wreath product k ∫ l → n (n = kl) and the ordinary product k × n−k→ n. We show that tr* preserves the form of these elements.

v 1 - and v 2 -Periodicity in Stable Homotopy Theory

1981 ◽  
Vol 103 (4) ◽  
pp. 615 ◽  
Author(s):  
Donald M. Davis ◽  
Mark Mahowald
Keyword(s):  
Homotopy Theory ◽  
Stable Homotopy ◽  
Stable Homotopy Theory

Spectra of derived module homomorphisms

1987 ◽  
Vol 101 (2) ◽  
pp. 249-257 ◽  
Author(s):  
Alan Robinson
Keyword(s):  
Spectral Sequence ◽  
Homotopy Theory ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Stable Homotopy Theory ◽  
Ring Spectrum ◽  
Homotopy Invariant ◽  
New Construction

We introduce a new construction in stable homotopy theory. If F and G are module spectra over a ring spectrum E, there is no well-known spectrum of E-module homomorphisms from F to G. Such a construction would not be homotopy invariant, and therefore would not serve much purpose. We show that, provided the rings and modules have A∞ structures, there is a spectrum RHomE(F, G) of derived module homomorphisms which has very pleasant properties. It is homotopy invariant, exact in each variable, and its homotopy groups form the abutment of a hypercohomology-type spectral sequence.

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