The Milnor Construction: Homotopy Classification of Principal Bundles

2007 ◽  
pp. 75-81
Author(s):  
D. Husemöller ◽  
M. Joachim ◽  
B. Jurčo ◽  
M. Schottenloher
Keyword(s):  
Homotopy Classification ◽  
Principal Bundles

scholarly journals Two-Categorical Bundles and their Classifying Spaces

2012 ◽  
Vol 10 (2) ◽  
pp. 299-369 ◽  
Author(s):  
Nils A. Baas ◽  
Marcel Bökstedt ◽  
Tore August Kro
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Vector Spaces ◽  
Bundle Theory ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Principal Bundles

AbstractFor a 2-category 2C we associate a notion of a principal 2C-bundle. For the 2-category of 2-vector spaces, in the sense of M.M. Kapranov and V.A. Voevodsky, this gives the 2-vector bundles of N.A. Baas, B.I. Dundas and J. Rognes. Our main result says that the geometric nerve of a good 2-category is a classifying space for the associated principal 2-bundles. In the process of proving this we develop powerful machinery which may be useful in further studies of 2-categorical topology. As a corollary we get a new proof of the classification of principal bundles. Another 2-category of 2-vector spaces has been proposed by J.C. Baez and A.S. Crans. A calculation using our main theorem shows that in this case the theory of principal 2-bundles splits, up to concordance, as two copies of ordinary vector bundle theory. When 2C is a cobordism type 2-category we get a new notion of cobordism-bundles which turns out to be classified by the Madsen–Weiss spaces.

scholarly journals Homotopy classification of elliptic operators on stratified manifolds

2006 ◽  
Vol 73 (3) ◽  
pp. 407-411 ◽  
Author(s):  
V. E. Nazaikinskii ◽  
A. Yu. Savin ◽  
B. Yu. Sternin
Keyword(s):  
Elliptic Operators ◽  
Homotopy Classification

On Graded Categorical Groups and Equivariant Group Extensions

2002 ◽  
Vol 54 (5) ◽  
pp. 970-997 ◽  
Author(s):  
A. M. Cegarra ◽  
J. M. Garćia-Calcines ◽  
J. A. Ortega
Keyword(s):  
Group Cohomology ◽  
Group Extension ◽  
Extension Problem ◽  
Homotopy Classification ◽  
Group Extensions ◽  
Categorical Groups

AbstractIn this article we state and prove precise theorems on the homotopy classification of graded categorical groups and their homomorphisms. The results use equivariant group cohomology, and they are applied to show a treatment of the general equivariant group extension problem.

scholarly journals Stable homotopy classification of BGp̌

Topology ◽  
1995 ◽  
Vol 34 (3) ◽  
pp. 633-649 ◽  
Author(s):  
John Martino ◽  
Stewart Priddy
Keyword(s):  
Stable Homotopy ◽  
Homotopy Classification

Unstable homotopy classification of

1996 ◽  
Vol 119 (1) ◽  
pp. 119-137 ◽  
Author(s):  
John Martino ◽  
Stewart Priddy
Keyword(s):  
Lie Group ◽  
Compact Lie Group ◽  
Classifying Space ◽  
Homotopy Classification ◽  
Completion Process

For nilpotent spaces p-completion is well behaved and reasonably well understood. By p–completion we mean Bousfield–Kan completion with respect to the field Fp [BK]. For non-nilpotent spaces the completion process often has a chaotic effect, this is true even for small spaces. One knows, however, that the classifying space of a compact Lie group is Fp-good even though it is usually non-nilpotent.

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