Borel Cohomology Research Articles

Seiberg–Witten Floer spectra and contact structures

In this article, the author defines an invariant of rational homology 3-spheres equipped with a contact structure as an element of a cohomotopy set of the Seiberg–Witten Floer spectrum as defined in Manolescu (Geometry Topol 7(2):889–932, 2003). Furthermore, in light of the equivalence established in Lidman and Manolescu (Astérisque 399:25, 2018) between the Borel equivariant homology of said spectrum and the Seiberg–Witten Floer homology of Kronheimer and Mrowka (Monopoles and three-manifolds, vol. 10, Cambridge University Press, Cambridge, 2007), the author shall show that this homotopy theoretic invariant recovers the already well known contact element in the Seiberg–Witten Floer cohomology (vid. e.g. Kronheimer et al. in Ann Math 20:457–546, 2007) in a natural fashion. Next, the behaviour of the cohomotopy invariant is considered in the presence of a finite covering. This setting naturally asks for the use of Borel cohomology equivariant with respect to the group of deck transformations. Hence, a new equivariant contact invariant is defined and its properties studied. The invariant is then computed in one concrete example, wherein the author demonstrates that it opens the possibility of considering scenarios hitherto inaccessible.

K-Theory and Formality

Abstract We compare several notions of equivariant formality for complex K-theory with respect to a compact Lie group action, including surjectivity of the forgetful map and the weak equivariant formality of Harada–Landweber, and find all are equivalent under standard hypotheses. As a consequence, we present an expression for the equivariant K-theory of the isotropy action of $H$ on a homogeneous space $G/H$ in all the classical cases. The proofs involve mainly homological algebra and arguments with the Atiyah–Hirzebruch–Leray–Serre spectral sequence, but a more general result depends on a map of spectral sequences from Hodgkin’s Künneth spectral sequence in equivariant K-theory to that in Borel cohomology that seems not to have been otherwise defined. The hypotheses for the main structure result are analogous to a previously announced characterization of cohomological equivariant formality, first proved here, expanding on the results of Shiga and Takahashi.

Khovanov homotopy type, periodic links and localizations

Given an m-periodic link Lsubset S^3, we show that the Khovanov spectrum mathcal {X}_L constructed by Lipshitz and Sarkar admits a group action. We relate the Borel cohomology of mathcal {X}_L to the equivariant Khovanov homology of L constructed by the second author. The action of Steenrod algebra on the cohomology of mathcal {X}_L gives an extra structure of the periodic link. Another consequence of our construction is an alternative proof of the localization formula for Khovanov homology, obtained first by Stoffregen and Zhang. By applying the Dwyer–Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link.

Rank conditions for finite group actions on 4-manifolds

AbstractLet M be a closed, connected, orientable topological $4$ -manifold, and G be a finite group acting topologically and locally linearly on M. In this paper, we investigate the spectral sequence for the Borel cohomology $H^*_G(M)$ and establish new bounds on the rank of G for homologically trivial actions with discrete singular set.

Looijenga line bundles in complex analytic elliptic cohomology

We present a calculation, which shows how the moduli of complex analytic elliptic curves arises naturally from the Borel cohomology of an extended moduli space of $U(1)$-bundles on a torus. Furthermore, we show how the analogous calculation, applied to a moduli space of principal bundles for a $K(\mathbb{Z},2)$ central extension of $U(1)^d$ give rise to Looijenga line bundles. We then speculate on the relation of these calculations to the construction of complex analytic equivariant elliptic cohomology.

Borel cohomology and the relative Gorenstein condition for classifying spaces of compact Lie groups

For a compact Lie group G we show that the representing spectrum for Borel cohomology generates its category of modules if G is connected or if coefficients are of characteristic p and π0(G) is a p-group. For a closed subgroup H of G we consider the map C⁎(BG)⟶C⁎(BH) and establish the sense in which it is relatively Gorenstein. Throughout, we pay careful attention to the importance of connectedness of the groups.

The FKMM-invariant in low dimension

In this paper we investigate the problem of the cohomological classification of "Quaternionic" vector bundles in low-dimension ($d\leqslant 3$). We show that there exists a characteristic classes $\kappa$, called the FKMM-invariant, which takes value in the relative equivariant Borel cohomology and completely classifies "Quaternionic" vector bundles in low-dimension. The main subject of the paper concerns a discussion about the surjectivity of $\kappa$.

A Non-trivial Ghost Kernel for Complex Projective Spaces with Symmetries

It is shown that the ghost kernel for certain equivariant stable cohomotopy groups of projective spaces is non-trivial. The proof is based on the Borel cohomology Adams spectral sequence and the calculations with the Steenrod algebra afforded by it.

The Borel cohomology of the loop space of a homogeneous space

Let B′→fB←pE 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 S1-equivariant cohomology of the space of the based loops on the complex projective space CPm, where the S1-action is induced by a linear action of S1 on CPm.

A geometric criterion for the boundedness of characteristic classes

We show that for a connected Lie group G, the linearity of its radical \({\sqrt G}\) (that is of its biggest connected normal solvable subgroup), is a necessary and sufficient condition for the boundedness of all Borel cohomology classes of G with integer coefficients, and that the linearity of \({\sqrt G}\) is also equivalent to a large-scale geometric property of G (involving distortion).

On the nonexistence of higher twistings over a point for Borel cohomology of [formula omitted]-theory

In this note we show that there are no higher twistings for the Borel cohomology theory associated to G -equivariant K -theory over a point and for a compact Lie group G . Therefore, twistings over a point for this theory are classified by the group H 1 ( B G , Z / 2 ) × H 3 ( B G , Z ) .

Generalized Artin and Brauer induction for compact Lie groups

Let G be a compact Lie group. We present two induction theorems for certain generalized G-equivariant cohomology theories. The theory applies to G-equivariant K-theory K G , and to the Borel cohomology associated with any complex oriented cohomology theory. The coefficient ring of K G is the representation ring R(G) of G. When G is a finite group the induction theorems for K G coincide with the classical Artin and Brauer induction theorems for R(G).

Coordinatewise decomposition, Borel cohomology, and invariant measures

Given Polish spaces X and Y and a Borel set S ⊆ X × Y with countable sections, we describe the circumstances under which a Borel function f : S → R is of the form f(x, y) = u(x) + v(y), where u : X → R and v : Y → R are Borel. This turns out to be a special case of the problem of determining whether a real-valued Borel cocycle on a countable Borel equivalence relation is a coboundary. We use several Glimm-Effros style dichotomies to give a solution to this problem in terms of certain σ-finite measures on the underlying space. The main new technical ingredient is a characterization of the existence of type III measures of a given cocycle. Suppose that S ⊆ X×Y and G is a group. A coordinatewise decomposition of a function f : S → G is a pair (u, v), where u : X → G, v : Y → G, and ∀(x, y) ∈ S (f(x, y) = u(x)v(y)). If X and Y are Polish spaces, G is a standard Borel group, and u and v are Borel, then we say that (u, v) is a Borel coordinatewise decomposition of f . Our main goal here is to show that when S is a Borel set with countable sections, f : S → G is Borel, and G = 〈R,+〉, the existence of a Borel coordinatewise decomposition can be characterized in terms of certain σ-finite measures on the disjoint union of X and Y (by a measure on a Polish space, we shall mean always a measure on its Borel subsets). Before getting to this, however, we consider first the existence of coordinatewise decompositions, without imposing any definability restrictions. ∗The author was supported in part by NSF VIGRE Grant DMS-0502315.

On the Borel cohomology of free loop spaces

Let $X$ be a space and let $K = H^*(X; \boldsymbol F_p)$ where $p$ is an odd prime. We construct functors $\overline \Omega$ and $\ell$ which approximate cohomology of the free loop space $\Lambda X$ as follows: There are homomorphisms $\overline \Omega(K) \to H^*(\Lambda X; \boldsymbol F_p)$ and $\ell(K)\to H^*(E\boldsymbol T\times_T\Lambda X;\boldsymbol F_p)$. These are isomorphisms when $X$ is a product of Eilenberg-MacLane spaces of type $K(\boldsymbol F_p,n)$ for $n \geq 1$.

Substitutional lemma for G-spaces of 1-dimensional groups

Let G be a compact Lie group and X a G-CW complex. We are interested in the calculation of the Borel cohomology of Xwhere EG is a universal free G-space and we use on the right hand side cellular cohomology. For an introduction to G-CW complexes see Matumoto [4] and for a good exposition on Borel cohomology see for instance torn Dieck [2], We want to replace X with an ordinary CW complex Y in order to find an ordinary CW structure on the Borel construction EG ΧGY so we can use cellular chains to compute the Borel cohomology of X. For every compact Lie group one has an extensionwhere G0 is the identity component, so for our case G0 is isomorphic to the circle group . We are dealing with the case in which π0(G) is isomorphic to C2, the cyclic group of order

Generalized Eilenberg–Moore spectral sequences for elementary abelian groups and tori

AbstractIn this note we prove universal coefficient theorems for Borel cohomology and related theories. Whatever other merit this may have the comment of Borel [5] applies ‘ …elle a au moms l'utilité de bien mettre en évidence le rôle fondamental joué dans cette question par la cohomologie des groupes’.Indeed the purpose of the enterprise is to use homological properties of the group cohomology ring H*(BG+) to study properties of G-spaces. Because of the relative simplicity of ordinary cohomology much attention in the proofs and applications is concentrated on change of groups, and on changes in the way the group action is exploited. Nonetheless we are able to adapt the non-equivariant approach of Adams ([1, 2]; see also [3]). Thus the existence of universal coefficient theorems automatically gives Kiinneth theorems as special cases.

Characteristic classes in borel cohomology

Let G be a topological group and let EC be a free contractible G-space. The Bore1 construction on a G-space X is the orbit space Xo = EC x G X. When asked what equivariant cohomology is, most people would answer Bore1 cohomology, namely H:(X) = H*(X,). This theory has the claim of priority and the merit of ready computability, and many very beautiful results have been proven with it. However, it suffers from the defects of its virtues. Precisely, it is ‘invariant’, in the sense that a G-map f: X + Y which is a nonequivariant homotopy equivalence induces an isomorphism on Bore1 cohomology. A quick way to see this is to observe that 1 x f: EC xX+ EC x Y is a map of principal G-bundles with base map lxof=fo:Xo-‘Y,, so that fc is a weak homotopy equivalence. As explained in [l], this invariance property is the crudest of a hierarchy of such properties that a theory might have. We shall show how to compute all characteristic classes in any invariant equivariant cohomology theory, the conclusion being that no such theory is powerful enough to support a very useful theory of characteristic classes. As explained in [2], a less crude invariance property can sometimes be exploited to obtain a calculation of equivariant characteristic classes in more powerful theories, such as equivariant K-theory. To establish context, consider a closed normal subgroup Z7 of a topological group r with quotient group G. If Y is a n-free r-space, we think of the orbit projection q : Y-* Y/Z7 as a particular kind of equivariant bundle. It is a principal U-bundle in the usual sense, its base space is a G-space and thus a Z7-trivial r-space, and its projection is a r-map. The classical case is r= G x Z7. Here q is called a principal (G, Ill)-bundle, and there is a theory of associated (G, I;T)-bundles exactly as in the nonequivariant case. For example, if G is a compact Lie group acting smoothly on a differentiable manifold M”, then the tangent n-plane bundle of M is a (G, O(n))bundle and is determined by its associated principal (G, O(n))-bundle. See e.g. [4; 5; 7, V $11 for background. There is a universal bundle E(Z7;r) -+ B(I7;I’) = E(Zl7; r)/Ii’ of the sort just specified. Its total space E(L7;T) is a Z7-free T-CW complex such that the /l-fixed point space E(Z7;r)” is contractible for any closed subgroup A of r such that

The lifting of an İnönü–Wigner contraction at the level of universal coverings

It is shown that, when the Borel cohomology of a connected Lie group G is such that all projective representations can be lifted to unitary representations of the universal covering group, then any contraction of G corresponds to a contraction of its universal covering. Three theorems are stated and proved. The results apply also to the İnönü–Wigner contraction of the Poincaré group into the Galilei group.

Differential characters and Borel cohomology

61. THE MAIN RESULTS THE PURPOSE of this note is to show that the differential characters associated to a compact Lie group G (as defined by Chern, Simons and Cheeger) comprise its Bore1 cohomology H&,,(G,R/Z). Along the way we obtain some interesting formulae for these characters as cochains in the bar resolution for G, and obtain some results on the cohomology of certain bundles with discrete structure group. Let % be the Lie algebra of G and let I(%, R) be the ring of Weil polynomials, i.e. symmetric multilinear functions on Yl which are invariant under the Adjoint action of G. In 121 Chern and Simons showed how to use a connection 8 on a principal G-bundle E-B to determine a characteristic form TP(8) in A*(E, R) for every P E I(%, R). These forms satisfy the equation

Cohomology for ergodic groupoids

Introduction. We construct a cohomology theory for the category of Mackey's ergodic groupoids and homomorphisms (but excluding certain homomorphisms considered by Mackey). Then various results of Mackey on ergodic groupoids and ergodic actions of groups are interpreted in the framework of this cohomology theory. Preliminary definitions regarding Borel groupoids are given in ?1. In ?2 we define cohomology groups and induced homomorphisms for Borel groupoids. This is seen to be a generalization of the cohomology theory for Borel groups as done by C. C. Moore in [4] (assuming the action on the coefficient group is trivial). Next, in ?3 we define the cohomology for ergodic groupoids, essentially by identifying Borel n-cochains which agree on some inessential contraction. An exact sequence relates the ergodic cohomology and the Borel cohomology for the underlying Borel groupoid. For an ergodic groupoid, .F C, and abelian ergodic coefficient group, A, we find HO(JF; A) A, and HI(JF; A) ={homomorphisms: Y -+ A} mod similarity (or equivalently, mod principal homomorphisms). If 0 and V are similar homomorphisms (as defined in this paper) .F -9 (ergodic groupoids) then we