scholarly journals Classifying spaces for commutativity of low-dimensional Lie groups

2019 ◽  
Vol 169 (3) ◽  
pp. 433-478 ◽  
Author(s):  
OMAR ANTOLÍN–CAMARENA ◽  
SIMON PHILIPP GRITSCHACHER ◽  
BERNARDO VILLARREAL
Keyword(s):  
Lie Groups ◽  
Homotopy Type ◽  
Steenrod Algebra ◽  
Homotopy Groups ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Cohomology Rings ◽  
Homotopy Fiber ◽  
Low Dimensional

AbstractFor each of the groups G = O(2), SU(2), U(2), we compute the integral and $\mathbb{F}_2$-cohomology rings of BcomG (the classifying space for commutativity of G), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of EcomG (the homotopy fiber of the inclusion BcomG → BG), and some low-dimensional homotopy groups of BcomG.

scholarly journals On the homotopy theory of monoids

1989 ◽  
Vol 47 (2) ◽  
pp. 171-185
Author(s):  
Carol M. Hurwitz
Keyword(s):  
Homotopy Type ◽  
Homotopy Theory ◽  
Small Category ◽  
Homotopy Equivalence ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Finitely Presented

AbsractIn this paper, it is shown that any connected, small category can be embedded in a semi-groupoid (a category in which there is at least one isomorphism between any two elements) in such a way that the embedding includes a homotopy equivalence of classifying spaces. This immediately gives a monoid whose classifying space is of the same homotopy type as that of the small category. This construction is essentially algorithmic, and furthermore, yields a finitely presented monoid whenever the small category is finitely presented. Some of these results are generalizations of ideas of McDuff.

scholarly journals Remark on a theorem of E.H. Brown

1973 ◽  
Vol 9 (1) ◽  
pp. 55-60 ◽  
Author(s):  
Michael D. Alder
Keyword(s):  
Homotopy Type ◽  
Full Subcategory ◽  
Topological Spaces ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Homotopy Classes ◽  
Continuous Maps ◽  
Homotopy Functors

The proliferation of classifying spaces in recent years owes much to the theorem of Edgar H. Brown, Jr on the representability of homotopy functors. Since the theorem only gives a representation for functors defined on the category of spaces having the homotopy type of a CW complex, there is some interest in finding conditions under which the domain category may be enlarged. It appears that a version of the theorem holds for any small full subcategory of Htp, the category of topological spaces and homotopy classes of continuous maps, but that the resulting classifying space is generally intractable.

Stable decompositions of classifying spaces of finite abelianp-groups

1988 ◽  
Vol 103 (3) ◽  
pp. 427-449 ◽  
Author(s):  
John C. Harris ◽  
Nicholas J. Kuhn
Keyword(s):  
Finite Group ◽  
Cohomology Theory ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Image Position ◽  
A Finite Group

LetBGbe the classifying space of a finite groupG. Consider the problem of finding astabledecompositionintoindecomposablewedge summands. Such a decomposition naturally splitsE*(BG), whereE* is any cohomology theory.

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 Spherical posets from commuting elements

2018 ◽  
Vol 21 (4) ◽  
pp. 593-628 ◽  
Author(s):  
Cihan Okay
Keyword(s):  
Homotopy Type ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Universal Cover ◽  
Homotopy Equivalent ◽  
Classifying Space ◽  
Partially Ordered ◽  
Abelian Subgroups

AbstractIn this paper, we study the homotopy type of the partially ordered set of left cosets of abelian subgroups in an extraspecial p-group. We prove that the universal cover of its nerve is homotopy equivalent to a wedge of r-spheres where {2r\geq 4} is the rank of its Frattini quotient. This determines the homotopy type of the universal cover of the classifying space of transitionally commutative bundles as introduced in [2].

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