Let F → X → B be a fibre bundle with structure group G, where B is (d − 1)-connected and of finite dimension, d 1. We prove that the strong L-S category of X is less than or equal to m+ dim B d , if F has a cone decomposition of length m under a compatibility condition with the action of G on F . This gives a consistent prospect to determine the L-S category of non-simply connected Lie groups. For example, we obtain cat(PU(n)) 3(n − 1) for all n 1, which might be best possible, since we have cat(PU(p r )) = 3(p r − 1) for any prime p and r 1. Similarly, we obtain the L-S category of SO(n) for n 9 and PO(8). We remark that all the above Lie groups satisfy the Ganea conjecture on L-S category. 2004 Elsevier B.V. All rights reserved. MSC: primary 55M30; secondary 22E20, 57N60
We give a cellular decomposition of the compact connected Lie group Spin (7). We also determine the L-S categories of Spin (7) and Spin (8).2000 Mathematics Subject Classification. Primary 55M30, Secondary 22E20, 57N60.
We determine the Lusternik-Schnirelmann category of real Stiefel manifolds V n,k and quaternionic Stiefel manifolds X n,k for n 2k which is equal to the cup-length of the mod 2 cohomology of V n,k and the integer cohomology of X n,k , respectively.
In the mid-1970's, Kono, Mimura and Shimada ([24], [25], [23]) have determined the mod p cohomology groups of BP U (3) and BF 4 for p = 3, and BE 6 and BE 7 for p = 2 using the Rothenberg-Steenrod spectral sequence ) has also told us that a twisted tensor product as an injective resolution is relevant to the study of the cohomology via the spectral sequence. Let G be a compact, connected simple Lie group and LG denote the loop group which is an infinite dimensional manifold consisting of all C ∞ -maps from the circle to G. Our interest here lies in computing the mod p cohomology of the classifying space BLG of the loop group LG. In order to compute those cohomologies, we give a DGA structure to the twisted tensor products by also perturbing the algebra structure of tensor product (A, 0) ⊗ (X A , d). More precisely, we have the following theorem. 2000 Mathematics Subject Classification. 55T22, 57T35, 55S05. Key words and phrases. loop groups, twisted tensor products, the Hochschild spectral sequence, the bar and cobar type Eilenberg-Moore spectral sequences, TV-models.
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