Borel Construction Research Articles

On multiplicative spectral sequences for nerves and the free loop spaces

On multiplicative spectral sequences for nerves and the free loop spaces

Homology stability for asymptotic monopole moduli spaces

We prove homological stability for two different flavours of asymptotic monopole moduli spaces, namely moduli spaces of framed Dirac monopoles and moduli spaces of ideal monopoles . The former are Gibbons–Manton torus bundles over configuration spaces whereas the latter are obtained from them by replacing each circle factor of the fibre with a monopole moduli space by the Borel construction. They form boundary hypersurfaces in a partial compactification of the classical monopole moduli spaces. Our results follow from a general homological stability result for configuration spaces equipped with non-local data.

Reflexive homology

Reflexive homology is the homology theory associated to the reflexive crossed simplicial group; one of the fundamental crossed simplicial groups. It is the most general way to extend Hochschild homology to detect an order-reversing involution. In this paper we study the relationship between reflexive homology and the $C_2$-equivariant homology of free loop spaces. We define reflexive homology in terms of functor homology. We give a bicomplex for computing reflexive homology together with some calculations, including the reflexive homology of a tensor algebra. We prove that the reflexive homology of a group algebra is isomorphic to the homology of the $C_2$-equivariant Borel construction on the free loop space of the classifying space. We give a direct sum decomposition of the reflexive homology of a group algebra indexed by conjugacy classes of group elements, where the summands are defined in terms of a reflexive analogue of group homology. We define a hyperhomology version of reflexive homology and use it to study the $C_2$-equivariant homology of certain free loop and free loop-suspension spaces. We show that reflexive homology satisfies Morita invariance. We prove that under nice conditions the involutive Hochschild homology studied by Braun and by Fernàndez-València and Giansiracusa coincides with reflexive homology.

T-equivariant disc potential and SYZ mirror construction

T-equivariant disc potential and SYZ mirror construction

Rational torus-equivariant stable homotopy V: The torsion Adams spectral sequence

Rational torus-equivariant stable homotopy V: The torsion Adams spectral sequence

Morse inequalities for orbifold Borel homology

Using a notion of Morse function on classifying spaces for finite groups, we define Morse numbers relating the critical point data of an orbifold Morse function to the homology of an associated Borel construction. In particular, we establish Morse inequalities relating our numbers to integral equivariant homology, generalizing the inequalities for manifold Morse functions due to E. Pitcher. To showcase the sharpness of these new inequalities, we will provide a complete classification of all closed, orientable, two-dimensional orbifolds for which equality may be achieved. For three-dimensional orbifolds with singular circles, we prove partial results in this direction, with an emphasis on singular knots in orbifold three-spheres.

Geometry of compact lifting spaces

We study a natural generalization of inverse systems of finite regular covering spaces. A limit of such a system is a fibration whose fibres are profinite topological groups. However, as shown in Conner et al. (Topol Appl 239:234–243, 2018), there are many fibrations whose fibres are profinite groups, which are far from being inverse limits of coverings. We characterize profinite fibrations among a large class of fibrations and relate the profinite topology on the fundamental group of the base with the action of the fundamental group on the fibre, and develop a version of the Borel construction for fibrations whose fibres are profinite groups.

Intertwining for semidirect product operads

This paper shows that the semi-direct product construction for $G$-operads and the levelwise Borel construction for $G$-cooperads are intertwined by the topological operadic bar construction. En route we give a generalization of the bar construction of M. Ching from reduced to certain non-reduced topological operads.

ON THE SYMMETRIC SQUARES OF COMPLEX AND QUATERNIONIC PROJECTIVE SPACE

AbstractThe problem of computing the integral cohomology ring of the symmetric square of a topological space has long been of interest, but limited progress has been made on the general case until recently. We offer a solution for the complex and quaternionic projective spaces$\mathbb{K}$Pn, by utilising their rich geometrical structure. Our description involves generators and relations, and our methods entail ideas from the literature of quantum chemistry, theoretical physics, and combinatorics. We begin with the case$\mathbb{K}$P∞, and then identify the truncation required for passage to finiten. The calculations rely upon a ladder of long exact cohomology sequences, which compares cofibrations associated to the diagonals of the symmetric square and the corresponding Borel construction. These incorporate the one-point compactifications of classic configuration spaces of unordered pairs of points in$\mathbb{K}$Pn, which are identified as Thom spaces by combining Löwdin's symmetric orthogonalisation (and its quaternionic analogue) with a dash ofPingeometry. The relations in the ensuing cohomology rings are conveniently expressed using generalised Fibonacci polynomials. Our conclusions are compatible with those of Gugnin mod torsion and Nakaoka mod 2, and with homological results of Milgram.

Mobile icosapods

Pods are mechanical devices constituted of two rigid bodies, the base and the platform, connected by a number of other rigid bodies, called legs, that are anchored via spherical joints. It is possible to prove that the maximal number of legs of a mobile pod, when finite, is 20. In 1904, Borel designed a technique to construct examples of such 20-pods, but could not constrain the legs to have base and platform points with real coordinates. We show that Borel's construction yields all mobile 20-pods, and that it is possible to construct examples with all real coordinates.

A model of the Borel construction on the free loopspace

Let $X$ be a CW-complex with basepoint. We obtain a simple description of the Borel construction on the free loopspace of the suspension of $X$ as a wedge of the classifying space of the circle and the homotopy colimit of a diagram consisting of products of a number of copies of $X$ and the standard topological $n$-simplex. This is obtained by filtering the cyclic bar construction on the James model of the based loopspace by word length in order to express the homotopy type of the free loopspace as a colimit of powers of $X$ and standard cyclic sets. It is shown that this colimit is in fact a homotopy colimit and commutativity of homotopy colimits is used to describe the Borel construction.

On the homotopy type of higher orbifolds and Haefliger classifying spaces

We describe various equivalent ways of associating to an orbifold, or more generally a higher étale differentiable stack, a weak homotopy type. Some of these ways extend to arbitrary higher stacks on the site of smooth manifolds, and we show that for a differentiable stack X arising from a Lie groupoid G , the weak homotopy type of X agrees with that of B G . Using this machinery, we are able to find new presentations for the weak homotopy type of certain classifying spaces. In particular, we give a new presentation for the Borel construction M × G E G of an almost free action of a Lie group G on a smooth manifold M as the classifying space of a category whose objects consist of smooth maps R n → M which are transverse to all the G -orbits, where n = dim ⁡ M − dim ⁡ G . We also prove a generalization of Segal's theorem, which presents the weak homotopy type of Haefliger's groupoid Γ q as the classifying space of the monoid of self-embeddings of R q , B ( Emb ( R q ) ) , and our generalization gives analogous presentations for the weak homotopy type of the Lie groupoids Γ 2 q S p and R Γ q which are related to the classification of foliations with transverse symplectic forms and transverse metrics respectively. We also give a short and simple proof of Segal's original theorem using our machinery.

Computational examples of rational string operations on Gorenstein spaces

In this paper, we give computational examples of string operations over the rational numbers field on Gorenstein spaces introduced by Félix and Thomas. Especially, we determine the structure of rational string operations on the classifying space of a compact connected Lie group and the Borel construction associated to an action of $S^{1}$ to $S^{2}$.

Simplicial monoid actions and associated monoid constructions

For \(X\) a pointed simplicial set with a simplicial monoid action, we construct a simplicial monoid and give a sufficient condition for its classifying space to be the homotopy cofiber of the inclusion of \(X\) into its reduced Borel construction. The known formulae for the respective classifying spaces of the four classical constructions of James, Milnor, Carlsson and Wu are special cases of this result. Thus, we unify categorially the four above-mentioned constructions. We also give the classical constructions and an example as the applications of the main result.

Homology decompositions of the loops on 1–stunted Borel constructions ofC2–actions

Carlsson's construction is a simplicial group whose geometric realization is the loop space of the 1-stunted reduced Borel construction. Our main results are: i) Given a pointed simplicial set acted upon by the discrete cyclic group C_2 of order 2, if the orbit projection has a section, then this loop space has a mod 2 homology decomposition; ii) If the reduced diagonal map of the C_2-invariant set is homologous to zero, then the pinched sets in the above homology decomposition themselves have homology decompositions in terms of the C_2-invariant set and the orbit space. Result i) generalizes a previous homology decomposition of the second author for trivial actions. To illustrate these two results, we completely compute the mod 2 Betti numbers for an example.

Support varieties for transporter category algebras

Abstract Let G be a finite group. Over any finite G -poset P we may define a transporter category as the corresponding Grothendieck construction. The classifying space of the transporter category is the Borel construction on the G -space B P , while the k -category algebra of the transporter category is the (Gorenstein) skew group algebra on the G -algebra k P . We introduce a support variety theory for the category algebras of transporter categories. It extends Carlson’s support variety theory on group cohomology rings to equivariant cohomology rings. In the mean time it provides a class of (usually non selfinjective) algebras to which Snashall–Solberg’s (Hochschild) support variety theory applies. Various properties will be developed. Particularly we establish a Quillen stratification for modules.

The Borel cohomology of the loop space of a homogeneous space

Let B ′ → f B ← p E be a diagram in which p is a fibration and the pair ( f , p ) of the maps is relatively formalizable. Then, we show that the rational cohomology algebra of the pullback of the diagram is isomorphic to the torsion product of algebras H ⁎ ( B ′ ) and H ⁎ ( E ) over H ⁎ ( B ) . Let M be a space which admits an action of a Lie group G . The isomorphism of algebras enables us to represent the cohomology of the Borel construction of the space of free (resp. based) loops on M in terms of the torsion product if M is equivariantly formal (resp. G -formal). Moreover, we compute explicitly the S 1 -equivariant cohomology of the space of the based loops on the complex projective space C P m , where the S 1 -action is induced by a linear action of S 1 on C P m .

An algebraic analog of the borel construction and its properties

Suppose that G is an affine algebraic group scheme faithfully flat over another affine scheme X = SpecR, H is a closed faithfully flat X-subscheme, and G/H is an affine X-scheme. In this case, we prove that the categories of left R[H]-comodules and G-equivariant vector bundles over G/H are equivalent and this equivalence respects tensor products. Our algebraic construction is based on a well-known geometric Borel construction. Bibliography: 5 titles.

Upper and Lower Bounds of the (co)chain Type Level of a Space

We establish an upper bound for the cochain type level of the total space of a pull-back fibration. It explains to us why the numerical invariants for principal bundles over the sphere are less than or equal to two. Moreover computational examples of the levels of path spaces and Borel constructions, including biquotient spaces and Davis-Januszkiewicz spaces, are presented. We also show that the chain type level of the homotopy fibre of a map is greater than the E-category in the sense of Kahl, which is an algebraic approximation of the Lusternik-Schnirelmann category of the map. The inequality fits between the grade and the projective dimension of the cohomology of the homotopy fibre.

Chern–Weil homomorphism in twisted equivariant cohomology

We describe the Cartan and Weil models of twisted equivariant cohomology together with the Cartan homomorphism among the two, and we extend the Chern–Weil homomorphism to the twisted equivariant cohomology. We clarify that in order to have a cohomology theory, the coefficients of the twisted equivariant cohomology must be taken in the completed polynomial algebra over the dual Lie algebra of G. We recall the relation between the equivariant cohomology of exact Courant algebroids and the twisted equivariant cohomology, and we show how to endow with a generalized complex structure the finite-dimensional approximations of the Borel construction M×GEGk, whenever the generalized complex manifold M possesses a Hamiltonian G-action.