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

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 On the K-theory of the loop space of a Lie group

1974 ◽  
Vol 76 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Francis Clarke
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Simple Structure ◽  
Homology Theory ◽  
Compact Lie Group ◽  
Classifying Space ◽  
K Theory ◽  
Simply Connected ◽  
Torsion Free ◽  
Multiplicative Structures

Let G be a simply connected, semi-simple, compact Lie group, let K* denote Z/2-graded, representable K-theory, and K* the corresponding homology theory. The K-theory of G and of its classifying space BG are well known, (8),(1). In contrast with ordinary cohomology, K*(G) and K*(BG) are torsion-free and have simple multiplicative structures. If ΩG denotes the space of loops on G, it seems natural to conjecture that K*(ΩG) should have, in some sense, a more simple structure than H*(ΩG).

On homotopy nilpotency

2021 ◽  
Vol 56 (2) ◽  
pp. 391-406
Author(s):  
Marek Golasiński ◽  
Keyword(s):  
Lie Group ◽  
Nilpotent Subgroup ◽  
Compact Lie Group ◽  
Loop Spaces

We review established and recent results on the homotopy nilpotence of spaces. In particular, the homotopy nilpotency of the loop spaces \(\Omega(G/K)\) of homogenous spaces \(G/K\) for a compact Lie group \(G\) and its closed homotopy nilpotent subgroup \(K \lt G\) is discussed.

Adjoint action of a finite loop space. II

1999 ◽  
Vol 129 (4) ◽  
pp. 773-785
Author(s):  
Norio Iwase ◽  
Akira Kono
Keyword(s):  
Lie Groups ◽  
Loop Space ◽  
Adjoint Action ◽  
Theoretic Approach ◽  
Loop Spaces ◽  
Classifying Space ◽  
Simply Connected ◽  
Prime 2 ◽  
Finite Loop

Adjoint actions of compact simply connected Lie groups are studied by Kozima and the second author based on the series of studies on the classification of simple Lie groups and their cohomologies. At odd primes, the first author showed that there is a homotopy theoretic approach that will prove the results of Kozima and the second author for any 1-connected finite loop spaces. In this paper, we use the rationalization of the classifying space to compute the adjoint actions and the cohomology of classifying spaces assuming torsion free hypothesis, at the prime 2. And, by using Browder's work on the Kudo–Araki operations Q1 for homotopy commutative Hopf spaces, we show the converse for general 1-connected finite loop spaces, at the prime 2. This can be done because the inclusion j: G > BAG satisfies the homotopy commutativity for any non-homotopy commutative loop space G.

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
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