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.

HOMOLOGY DECOMPOSITIONS FOR CLASSIFYING SPACES OF FINITE GROUPS ASSOCIATED TO MODULAR REPRESENTATIONS

2001 ◽  
Vol 64 (2) ◽  
pp. 472-488 ◽  
Author(s):  
D. NOTBOHM
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Homotopy Theory ◽  
Modular Representation ◽  
Modular Representations ◽  
Classifying Spaces ◽  
Modular Representation Theory ◽  
Classifying Space ◽  
Homotopy Colimit ◽  
A Finite Group

For a prime p, a homology decomposition of the classifying space BG of a finite group G consist of a functor F : D → spaces from a small category into the category of spaces and a map hocolim F → BG from the homotopy colimit to BG that induces an isomorphism in mod-p homology. Associated to a modular representation G → Gl(n; [ ]p), a family of subgroups is constructed that is closed under conjugation, which gives rise to three different homology decompositions, the so-called subgroup, centralizer and normalizer decompositions. For an action of G on an [ ]p-vector space V, this collection consists of all subgroups of G with nontrivial p-Sylow subgroup which fix nontrivial (proper) subspaces of V pointwise. These decomposition formulas connect the modular representation theory of G with the homotopy theory of BG.

The Morava K-theories of wreath products

1990 ◽  
Vol 107 (2) ◽  
pp. 309-318 ◽  
Author(s):  
John Hunton
Keyword(s):  
Homotopy Theory ◽  
Wreath Products ◽  
Stable Homotopy ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Stable Homotopy Theory ◽  
Recent Developments ◽  
Positive Integers ◽  
Extended Power ◽  
A Current

In p-primary stable homotopy theory, recent developments have shown the importance of the Morava K-theory spectra K(n) for positive integers n. A current major problem concerns the behaviour of the K(n)-cohomologies on the classifying spaces of finite groups and on related spaces. In this paper we show how to compute the Morava K-theory of extended power constructions Here Xp is the p-fold product of some space X and Cp is the cyclic group of order p. In particular, if we take X as the classifying space BG for some group G, then Dp(X) forms the classifying space for , the wreath product of G by Cp.

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 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.

Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

2013 ◽  
Author(s):  
Friedhelm Waldhausen ◽  
Bjørn Jahren ◽  
John Rognes
Keyword(s):  
Main Tool ◽  
Homotopy Equivalence ◽  
Classifying Space ◽  
Deformation Retract ◽  
Simplicial Sets ◽  
Full Proof

Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.

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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