scholarly journals Finite loop space with maximal tori have finite Weyl groups

1993 ◽  
Vol 119 (1) ◽  
pp. 299-299
Author(s):  
Larry Smith
Keyword(s):  
Loop Space ◽  
Weyl Groups ◽  
Maximal Tori ◽  
Finite Loop ◽  
Finite Loop Space

scholarly journals RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

2021 ◽  
pp. 1-29
Author(s):  
DREW HEARD
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Algebraic Model ◽  
Compact Lie Group ◽  
Loop Spaces ◽  
Local Systems ◽  
Special Case ◽  
Finite Loop ◽  
Finite Loop Space

Abstract Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group W G K is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

scholarly journals A finite loop space not rationally equivalent to a compact Lie group

2004 ◽  
Vol 157 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Kasper K. S. Andersen ◽  
Tilman Bauer ◽  
Jesper Grodal ◽  
Erik Kjaer Pedersen
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Compact Lie Group ◽  
Finite Loop ◽  
Finite Loop Space

scholarly journals The type of a torsion free finite loop space

1991 ◽  
Vol 42 (2) ◽  
pp. 175-186 ◽  
Author(s):  
James P. Lin ◽  
Frank Williams
Keyword(s):  
Loop Space ◽  
Torsion Free ◽  
Finite Loop ◽  
Finite Loop Space

Universal phantom maps out of loop spaces

2000 ◽  
Vol 130 (2) ◽  
pp. 313-333
Author(s):  
K. Iriye
Keyword(s):  
Loop Space ◽  
Compact Lie Group ◽  
Necessary And Sufficient Condition ◽  
Loop Spaces ◽  
Phantom Maps ◽  
Cw Complex ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Finite Loop ◽  
Finite Loop Space

We consider the universal phantom map out of a non-finite loop space. First we obtain a necessary and sufficient condition for the universal phantom map out of ΩG for a simply connected compact Lie group G to be essential. Next we prove that the universal phantom map out of ΩkX is essential for all k ≥ 2 if X is a simply connected non-contractible finite CW-complex. Ingredients in the proof are the Browder's ∞-implication argument and the Eilenberg–Moore spectral sequence.

scholarly journals A new finite loop space at the prime two

1993 ◽  
Vol 6 (1) ◽  
pp. 37-37 ◽  
Author(s):  
W. G. Dwyer ◽  
C. W. Wilkerson
Keyword(s):  
Loop Space ◽  
Finite Loop ◽  
Finite Loop Space

ON THE FIXED-POINT SETS OF TORUS ACTIONS ON FLAG MANIFOLDS

2002 ◽  
Vol 01 (03) ◽  
pp. 255-265 ◽  
Author(s):  
BARBARA A. SHIPMAN
Keyword(s):  
Fixed Point ◽  
Toda Lattice ◽  
Flag Manifold ◽  
Weyl Groups ◽  
Flag Manifolds ◽  
Unipotent Group ◽  
Point Sets ◽  
Torus Actions ◽  
Maximal Tori ◽  
Fixed Point Sets

This paper takes a detailed look at a subject that occurs in various contexts in mathematics, the fixed-point sets of torus actions on flag manifolds, and considers it from the (perhaps nontraditional) perspective of moment maps and length functions on Weyl groups. The approach comes from earlier work of the author where it is shown that certain singular flows in the Hamiltonian system known as the Toda lattice generate the action of a group A on a flag manifold, where A is a direct product of a non-maximal torus and unipotent group. As a first step in understanding the orbits of A in connection with the Toda lattice, this paper seeks to understand the fixed points of the non-maximal tori in this setting.

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