In this paper we characterize the fiber representations of equivariant complex vector bundles over a circle and classify these bundles. We also treat the triviality of equivariant complex vector bundles over a circle by investigating the extensions of representations. As a corollary of our results, we calculate the reduced equivariant K-group of a circle for any compact Lie group.
We consider Γ-equivariant principal G-bundles over proper Γ-CW-complexes with prescribed family of local representations. We construct and analyze their classifying spaces for locally compact, second countable topological groups with finite covering dimension Γand G such that G is almost connected.
For a space with involutive action, there is a variant of K-theory. Motivated by T-duality in type II orbifold string theory, we establish that a twisted version of the variant enjoys a topological T-duality for Real circle bundles, i.e. circle bundles with real structure.
We define equivariant projective unitary stable bundles as the appropriate twists when defining K-theory as sections of bundles with fibers the space of Fredholm operators over a Hilbert space. We construct universal equivariant projective unitary stable bundles for the orbit types, and we use a specific model for these local universal spaces in order to glue them to obtain a universal equivariant projective unitary stable bundle for discrete and proper actions. We determine the homotopy type of the universal equivariant projective unitary stable bundle, and we show that the isomorphism classe…
The classification of Seifert manifolds was given in terms of numeric data by Seifert in 1933, and then generalized by Orlik and Raymond in 1968 to circle actions on closed 3d manifolds. In this paper, we further generalize the classification to circle actions on 3d manifolds with boundaries by adding a numeric parameter and a union of cycle graphs. Then we describe the equivariant cohomology of 3d manifolds with circle actions in terms of ring, module and vector-space structures. We also compute equivariant Betti numbers and Poincaré series for these manifolds and discuss the equivariant form…
We give an explicit decription for univeral principal U(r)-bundles with periodic map by means of equivariant Stiefel manifolds. We then show that the associated equivariant vector bundle is equivalent to the canonical one given by G. Segal. Finally, we investigate some ideals involved in the equivariant K-theory of this classifying space.
We compare two notions of $G$-fiber bundles and $G$-principal bundles in the literature, with an aim to clarify early results in equivariant bundle theory that are needed in current work of equivariant algebraic topology. We also give proofs of some equivariant generalizations of well-known non-equivariant results involving the classifying space.
A generalisation of the equivariant Dixmier-Douady invariant is constructed as a second-degree cohomology class within a new semi-equivariant Čech cohomology theory. This invariant obstructs liftings of semi-equivariant principal bundles that are associated to central exact sequences of structure groups in which each structure group is acted on by the equivariance group. The results and methods described can be applied to the study of complex vector bundles equipped with linear/anti-linear actions, such as Atiyah's Real vector bundles.
The circle-equivariant spectrum MString_C is the equivariant analogue of the cobordism spectrum MU<6> of stably almost complex manifolds with c_1=c_2=0. Given a rational elliptic curve C, the second author has defined a ring T-spectrum EC representing the associated T-equivariant elliptic cohomology. The core of the present paper is the construction, when C is a complex elliptic curve, of a map of ring T-spectra MString_C --> EC which is the rational equivariant analogue of the sigma orientation of Ando-Hopkins-Strickland. We support this by a theory of characteristic classes for calculation, …
We study the fixed points of the universal G-equivariant n-dimensional complex vector bundle and obtain a decomposition formula in terms of twisted equivariant universal complex vector bundles of smaller dimension. We use this decomposition to describe the fixed points of the complex equivariant K-theory spectrum and the equivariant unitary bordism groups for adjacent families of subgroups.
In this paper, we prove that the orbit space B and the Euler class of an action of the circle S^1 on X determine both the equivariant intersection cohomology of the pseudomanifold X and its localization. We also construct a spectral sequence converging to the equivariant intersection cohomology of X whose third term is described in terms of the intersection cohomology of B.
We construct a canonical Thom isomorphism in Grojnowski's equivariant elliptic cohomology, for virtual T-oriented T-equivariant spin bundles with vanishing Borel-equivariant second Chern class, which is natural under pull-back of vector bundles and exponential under Whitney sum. It extends in the complex-analytic case the non-equivariant sigma orientation of Hopkins, Strickland, and the author. The construction relates the sigma orientation to the representation theory of loop groups and Looijenga's weighted projective space, and sheds light even on the non-equivariant case. Rigidity theorems …
We classify equivariant topological complex vector bundles over real projective plane under a compact Lie group (not necessarily effective) action. It is shown that nonequivariant Chern classes and isotropy representations at (at most) three points are sufficient to classify equivariant vector bundles over real projective plane except one case. To do it, we relate the problem to classification on two-sphere through the covering map because equivariant vector bundles over two-sphere have been already classified.
We exhaustively classify topological equivariant complex vector bundles over two-torus under a compact Lie group (not necessarily effective) action. It is shown that inequivariant Chern classes and isotropy representations at (at most) six points are sufficient to classify equivariant vector bundles except a few cases. To do it, we calculate homotopy of the set of equivariant clutching maps. And, classification on real projective plane, Klein bottle will appear soon
We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization. Beside explicit constructions of the metrics, this is achieved by identifying equivariant structures upon these vector bundles via a comparison of their equivariant and non-equivariant K-theory. For this, in particular, we transcribe equivariant K-theory to equivariant rational cohomology and investigate surjectivity properties of induced maps in the Borel fibration via rational homotopy theory…
We generalize the results of a previous paper of ours to compact Lie groups. Using a recently developed ordinary equivariant homology and cohomology, we define equivariant Poincare complexes with the properties that (1) every compact G-manifold is an equivariant Poincare complex, (2) every finite equivariant Poincare complex (with some mild additional hypotheses) has an equivariant spherical Spivak normal fibration, and (3) the Pi-Pi Theorem holds for equivariant Poincare pairs under suitable gap hypotheses. The nice behavior of the ordinary equivariant homology and cohomology theories allows …
We classify SO(n)-equivariant principal bundles over $S^n$ in terms of their isotropy representations over the north and south poles. This is an example of a general result classifying equivariant $(Π, G)$-bundles over cohomogeneity one manifolds.
Let $G$ be an infinite discrete group and let $\underline{E}G$ be a classifying space for proper actions of $G$. Every $G$-equivariant vector bundle over $\underline{E}G$ gives rise to a compatible collection of representations of the finite subgroups of $G$. We give the first examples of groups $G$ with a cocompact classifying space for proper actions $\underline{E}G$ admitting a compatible collection of representations of the finite subgroups of $G$ that does not come from a $G$-equivariant (virtual) vector bundle over $\underline{E}G$. This implies that the Atiyah-Hirzeburch spectral sequen…
We compute rational $T$-equivariant elliptic cohomology of CP(V), where $T$ is the circle group, and CP(V) is the $T$-space of complex lines for a finite dimensional complex $T$-representation V. Starting from an elliptic curve C over the complex numbers and a coordinate data around the identity, we achieve this computation by proving that the $T$-equivariant elliptic cohomology theory $EC_T$ built in [Gre05], and the $T^2$-equivariant elliptic cohomology theory $EC_{T^2}$ built in [Bar22] are 1x$T$-split. This result allows us to reduce the computation of $EC_T(CP(V))$ to the computation of $…
We relate endotrivial representations of a finite group in characteristic p to equivariant line bundles on the simplicial complex of non-trivial p-subgroups, by means of weak homomorphisms.
The main purpose this work is to study the minimal fixed point set of fiber-preserving maps for spaces which are fiber bundles over the circle and the fiber is the torus. Using the one-parameter fixed point theory is possible to describe these sets in terms of the fundamental group and the induced homomorphism.
We proof that having boundary of standard 3-dimensional simplex $\partial Δ^3$ as a base of triangulation one can triangulate only trivial and Hopf circle bundles.
We introduce a new construction, the isotropy groupoid, to organize the orbit data for split $Γ$-spaces. We show that equivariant principal $G$-bundles over split $Γ$-CW complexes $X$ can be effectively classified by means of representations of their isotropy groupoids. For instance, if the quotient complex $A=Γ\backslash X$ is a graph, with all edge stabilizers toral subgroups of $Γ$, we obtain a purely combinatorial classification of bundles with structural group $G$ a compact connected Lie group. If $G$ is abelian, our approach gives combinatorial and geometric descriptions of some results …
We study the existence of $S^1$-equivariant characteristic classes on certain natural infinite rank bundles over the loop space $LM$ of a manifold $M$. We discuss the different $S^1$-equivariant cohomology theories in the literature and clarify their relationships. We attempt to use $S^1$-equivariant Chern-Weil techniques to construct $S^1$-equivariant characteristic classes. The main result is the construction of a sequence of $S^1$-equivariant characteristic classes on the total space of the bundles, but these classes do not descend to the base $LM$. Nevertheless, we conclude by identifying …
For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the Cartan-Weil equivariant forms and to Borel's equivariant integral cohomology. We show the Chern-Weil homomorphism for equivariant vector bundles with connection naturally factors through differential equivariant cohomology.
In this note, we consider a Lie group G acting on a manifold M. We prove that the category of bundles with connection on the differential quotient stack is equivalent to the category of G-equivariant bundles on M with G-invariant connection.
We investigate a PL topology question: which circle bundles can be triangulated over a given triangulation of the base? The question got a simple answer emphasizing the role of minimal triangulations encoded by local systems of circular permutations of vertices of the base simplices. The answer is based on an experimental fact: classical Huntington transitivity axiom for cyclic orders can be expressed as the universal binary Chern cocycle.
Principal circle bundle over a PL polyhedron can be triangulated and thus obtains combinatorics. The triangulation is assembled from triangulated circle bundles over simplices. To every triangulated circle bundle over a simplex we associate a necklace (in combinatorial sense). We express rational local formulas for all powers of first Chern class in the terms of mathematical expectations of parities of the associated necklaces. This rational parity is a combinatorial isomorphism invariant of triangulated circle bundle over simplex, measuring mixing by triangulation of the circular graphs over …
Assume $(M, ω)$ is a connected, compact 6 dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action, such that the fixed point set consists of isolated points or compact orientable surfaces. We restrict attention to the case $\dim$ $H^2(M)<3$. We give a complete list of the possible manifolds, determine their equivariant cohomology ring and equivariant Chern classes. Some of these manifolds are classified up to diffeomorphism. We also show the existence for a few cases.
Let $M$ be an irreducible projective variety over an algebraically closed field $k$ of characteristic zero equipped with an action of a group $Γ$. Let $E_G$ be a principal $G$--bundle over $M$, where $G$ is a connected reductive algebraic group over $k$, equipped with a lift of the action of $Γ$ on $M$. We give conditions for $E_G$ to admit a $Γ$--equivariant reduction of structure group to $H$, where $H \subset G$ is a Levi subgroup. We show that for $E_G$, there is a naturally associated conjugacy class of Levi subgroups of $G$. Given a Levi subgroup $H$ in this conjugacy class, $E_G$ admits…
We define an elliptic version of the stable envelope of Maulik and Okounkov for the equivariant elliptic cohomology of cotangent bundles of Grassmannians. It is a version of the construction proposed by Aganagic and Okounkov and is based on weight functions and shuffle products. We construct an action of the dynamical elliptic quantum group associated with $\mathfrak{gl}_2$ on the equivariant elliptic cohomology of the union of cotangent bundles of Grassmannians. The generators of the elliptic quantum groups act as difference operators on sections of admissible bundles, a notion introduced in …
This paper generalizes Bismut's equivariant Chern character to the setting of abelian gerbes. In particular, associated to an abelian gerbe with connection, an equivariantly closed differential form is constructed on the space of maps of a torus into the manifold. These constructions are made explicit using a new local version of the higher Hochschild complex, resulting in differential forms given by iterated integrals. Connections to two dimensional topological field theories are indicated. Similarly, this local higher Hochschild complex is used to calculate the 2-holonomy of an abelian gerbe…
