## What is/are Classifying Space?

Classifying Space - These results are obtained by constructing CW approximations to the classifying spaces of these groups, in the category of E2$E_{2}$-algebras, which have no E2$E_{2}$-cells below a certain vanishing line.^{[1]}The classifying spaces of cobordisms of singular maps have two fairly different constructions.

^{[2]}As a direct consequence, we obtain a discrete analogue of cell decompositions in smooth Morse theory, by using the classifying space model introduced in Nanda et al (Discrete Morse theory and classifying spaces, arXiv:1612.

^{[3]}We study stably rationality and retract rationality properties of the classifying spaces of split spinor groups Spinn over a field F of characteristic not 2.

^{[4]}More recently, they have surfaced (sometimes unexpectedly) in various other contexts from free probability to classifying spaces of braid groups.

^{[5]}Castellana, Greenlees, and Grodal did some computations of the singularity and cosingularity categories of the ring of cochains on a classifying space of a group, to get a feeling for some unexplored structural properties.

^{[6]}As an application, we show that the classifying space of the τ-cluster morphism category of a τ-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) is an Eilenberg-MacLane space if every vertex in the corresponding quiver has degree at most 2.

^{[7]}We show that there is an example of a space for which the homotopy set is a non-commutative group, and hence the classifying space of the automorphism group of the Cuntz algebra for finite n is not an H-space.

^{[8]}We give an explicit simple construction for classifying spaces of maps obtained as hyperplane projections of immersions.

^{[9]}We also derive various spectral sequences for the equivariant cohomology of a differentiable stack generalising among others Bott's spectral sequence which converges to the cohomology of the classifying space of a Lie group.

^{[10]}We prove that if G is homotopy commutative then the homotopy type of the classifying space BG∗(P ) can be completely determined for certain M.

^{[11]}Finally in the last part of the paper we prove that under suitable assumptions on $V$ the push-forward of $[\overline{\eth}_{\mathrm{rel}}]$ in the K-homology of the classifying space of the fundamental group of $V$ is a birational invariant.

^{[12]}We present an alternative proof of this fact, using classifying spaces of families of subgroups.

^{[13]}We compute the integral cohomology ring of the classifying spaces of gauge groups of principal U(n)-bundles over the 2-sphere by generalizing the operation for free loop spaces, called the free double suspension.

^{[14]}The functors of classifying spaces and face posets are compatible with these homotopy theories.

^{[15]}By mapping the non-Hermitian system into an enlarged Hermitian Hamiltonian with an enforced chiral symmetry, our topological classification is thus equivalent to classifying Hermitian systems with both chiral and reflection symmetries, which effectively change the classifying space and shift the periodical table of topological phases.

^{[16]}The resulting space is a classifying space in the sense of algebraic topology.

^{[17]}It follows that the classifying space BG is p-retract rational if and only if there is a p-versal G-torsor Y → X with X a rational variety, that is, all G-torsors over infinite fields are rationally parameterized.

^{[18]}If a closed 3-manifold is considered as a 3D universe, then every 4D spacetime is embedded in every 4D universe and hence every 4D universe is a classifying space for every spacetime.

^{[19]}We prove that for any homotopy type $X$, there is an abstract elementary class $\mathcal{C}$, with joint embedding, almagamation and no maximal models such that the classifying space realizes the homotopy type $X$.

^{[20]}We prove a lemma about their fixed points when $G$ is a $p$-group, and then use this lemma to compute the fixed points of the Steinberg summand of the equivariant classifying space of $(\mathbb{Z}/p)^n$.

^{[21]}In particular, we show that if G is an amenable and connective discrete group whose classifying space BG is homotopic to a finite simplicial complex, then G is asymptotically stable.

^{[22]}We introduce classifying spaces for families of subgroups and deduce their homotopy theoretic characterization.

^{[23]}We prove the $K(\pi,1)$ conjecture for affine Artin groups: the complexified complement of an affine reflection arrangement is a classifying space.

^{[24]}We exploit the close connection between homology groups of $$\mathrm {GL}_n(R)$$GLn(R) for $$n\le 5$$n≤5 and those of related classifying spaces, then compute the former using Voronoi’s reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which $$\mathrm {GL}_n(R)$$GLn(R) acts.

^{[25]}On the other hand, the Chow ring of the classifying space of G over any field of characteristic ̸= 2 is known to contain non-zero torsion elements.

^{[26]}The definitions of space states are given, criteria for classifying space states into different levels and types are proposed, considering the areas and scope of space activities, and their various forms are shown.

^{[27]}These theories can be constructed as sigma models with target space the second classifying space B2G of the symmetry group G, and they are classified by cohomology classes of B2G.

^{[28]}We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space $\mathrm{Baut}_1(X_{\mathbb Q})$ as a consequence.

^{[29]}

## classifying space bg

We prove that if G is homotopy commutative then the homotopy type of the classifying space BG∗(P ) can be completely determined for certain M.^{[1]}It follows that the classifying space BG is p-retract rational if and only if there is a p-versal G-torsor Y → X with X a rational variety, that is, all G-torsors over infinite fields are rationally parameterized.

^{[2]}In particular, we show that if G is an amenable and connective discrete group whose classifying space BG is homotopic to a finite simplicial complex, then G is asymptotically stable.

^{[3]}