Equivariant Bordism Research Articles

Oriented and unitary equivariant bordism of surfaces

Oriented and unitary equivariant bordism of surfaces

The local (co)homology theorems for equivariant bordism

The local (co)homology theorems for equivariant bordism

Chern classes in equivariant bordism

Abstract We introduce Chern classes in $U(m)$ -equivariant homotopical bordism that refine the Conner–Floyd–Chern classes in the $\mathbf {MU}$ -cohomology of $B U(m)$ . For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees–May local homology spectral sequence collapses for products of unitary groups. We use the Chern classes to reprove the $\mathbf {MU}$ -completion theorem of Greenlees–May and La Vecchia.

Global group laws and equivariant bordism rings

For every abelian compact Lie group $A$, we prove that the homotopical $A$-equivariant complex bordism ring, introduced by tom Dieck (1970), is isomorphic to the $A$-equivariant Lazard ring, introduced by Cole--Greenlees--Kriz (2000). This settles a conjecture of Greenlees. We also show an analog for homotopical real bordism rings over elementary abelian $2$-groups. Our results generalize classical theorems of Quillen (1969) on the connection between non-equivariant bordism rings and formal group laws, and extend the case $A=C_2$ due to Hanke--Wiemeler (2018). We work in the framework of global homotopy theory, which is essential for our proof. In addition to the statements for a fixed group $A$, we also prove a global algebraic universal property that characterizes the collection of all equivariant complex bordism rings simultaneously. We show that they form the universal contravariant functor from abelian compact Lie groups to commutative rings that is equipped with a coordinate; the coordinate is given by the universal Euler class at the circle group. More generally, the ring of $n$-fold cooperations of equivariant complex bordism is shown to be universal among functors equipped with a strict $n$-tuple of coordinates.

Algebraic realization of actions of some finite groups

Let G be $$A_5$$ , $$A_4$$ , or a finite group with cyclic Sylow 2 subgroup. We show that every closed smooth G manifold M has a strongly algebraic model. This means, there exist a nonsingular real algebraic G variety X which is equivariantly diffeomorphic to M and all G vector bundles over X are strongly algebraic. Making use of improved blow-up techniques and the literature on equivariant bordism theory, we are extending older algebraic realization results.

Bordism of elementary abelian groups via inessential Brown–Peterson homology

We compute the equivariant bordism of free oriented $(\mathbb{Z}/p)^n$-manifolds as a module over $\Omega_*^{SO}$, when $p$ is an odd prime. We show, among others, that this module is canonically isomorphic to a direct sum of suspensions of multiple tensor products of $\Omega^{SO}_*(B \mathbb{Z}/p)$, and that it is generated by products of standard lens spaces. This considerably improves previous calculations by various authors. Our approach relies on the investigation of the submodule of the Brown-Peterson homology of $B (\mathbb{Z}/p)^n$ generated by elements coming from proper subgroups of $(\mathbb{Z}/p)^n$. We apply our results to the Gromov-Lawson-Rosenberg conjecture for atoral manifolds whose fundamental groups are elementary abelian of odd order.

Circle actions and scalar curvature

We construct metrics of positive scalar curvature on manifolds with circle actions. One of our main results is that there exist $S^1$-invariant metrics of positive scalar curvature on every $S^1$-manifold which has a fixed point component of codimension 2. As a consequence we can prove that there are non-invariant metrics of positive scalar curvature on many manifolds with circle actions. Results from equivariant bordism allow us to show that there is an invariant metric of positive scalar curvature on the connected sum of two copies of a simply connected semi-free $S^1$-manifold $M$ of dimension at least six provided that $M$ is not $\text{spin}$ or that $M$ is $\text{spin}$ and the $S^1$-action is of odd type. If $M$ is spin and the $S^1$-action of even type then there is a $k>0$ such that the equivariant connected sum of $2^k$ copies of $M$ admits an invariant metric of positive scalar curvature if and only if a generalized $\hat{A}$-genus of $M/S^1$ vanishes.

Torus manifolds in equivariant complex bordism

We restrict geometric tangential equivariant complex Tn-bordism to torus manifolds and provide a complete combinatorial description of the appropriate non-commutative ring. We discover, using equivariant K-theory characteristic numbers, that the information encoded in the oriented torus graph associated to a stably complex torus manifold completely describes its equivariant bordism class. We also consider the role of omnioriented quasitoric manifolds in this description.

Involutions with fixed point set RP(2m) ⊔ P(2m, 2n + 1)

Let (M, T) be a smooth closed manifold with a smooth involution T whose fixed point set is a disjoint union of an even-dimensional real projective space and a Dold manifold. In some cases, the equivariant bordism classes of (M, T) are determined.

Small Covers and the Equivariant Bordism Classification of 2-torus Manifolds

Associated with the Davis-Januszkiewicz theory of small covers, this paper deals with the theory of 2-torus manifolds from the viewpoint of equivariant bordism. We define a differential operator on the "dual" algebra of the unoriented $G_n$-representation algebra introduced by Conner and Floyd, where $G_n=(\Z_2)^n$. With the help of $G_n$-colored graphs (or mod 2 GKM graphs), we may use this differential operator to give a very simple description of tom Dieck-Kosniowski-Stong localization theorem in the setting of 2-torus manifolds. We then apply this to study the $G_n$-equivariant unoriented bordism classification of $n$-dimensional 2-torus manifolds. We show that the $G_n$-equivariant unoriented bordism class of each $n$-dimensional 2-torus manifold contains an $n$-dimensional small cover as its representative, solving the conjecture posed in [19]. In addition, we also obtain that the graded noncommutative ring formed by the equivariant unoriented bordism classes of 2-torus manifolds of all possible dimensions is generated by the classes of all generalized real Bott manifolds (as special small covers over the products of simplices). This gives a strong connection between the computation of $G_n$-equivariant bordism groups or ring and the Davis-Januszkiewicz theory of small covers. As a computational application, with the help of computer, we completely determine the structure of the group formed by equivariant bordism classes of all 4-dimensional 2-torus manifolds. Finally, we give some essential relationships among 2-torus manifolds, coloring polynomials, colored simple convex polytopes, colored graphs.

Real equivariant bordism for elementary abelian 2-groups

We give a description of real equivariant bordism for elementary abelian 2-groups, which is similar to the description of complex equivariant bordism for the group S^1 x ... x S^1 given by Hanke in 2005.

The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order

We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group \({\mathbb{Z}_p}\), where p is an odd prime. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p = n = 3. We use the expressions obtained to show that any such manifold is trivial in the appropriate reduced equivariant spin bordism group.

Positive scalar curvature with symmetry

We show an equivariant bordism principle for constructing metrics of positive scalar curvature that are invariant under a given group action. Furthermore, we develop a new codimension-2 surgery technique which removes singular strata from fixed point free S1-manifolds while preserving equivariant positive scalar curvature. These results are applied to derive the following theorem: Each closed fixed point free S1-manifold of dimension at least 6 whose isotropy groups have odd order and whose union of maximal orbits is simply connected and not spin, carries an S1-invariant metric of positive scalar curvature.

Geometric versus homotopy theoretic equivariant bordism

By results of Loffler and Comezana, the Pontrjagin-Thom map from geometric G-equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian G. If G=S1×⋯×S1, we prove that the associated fixed point square is a pull back square, thus confirming a recent conjecture of Sinha [22]. This is used in order to determine the image of the Pontrjagin-Thom map for toralG.

Contact structures, equivariant spin bordism, and periodic fundamental groups

Using contact surgery and equivariant bordism theory, we prove the existence of contact structures on all 5-dimensional spin manifolds with certain finite fundamental groups.

Real equivariant bordism and stable transversality obstructions for $\mathbb {Z}/2$

In this paper we compute homotopical equivariant bordism for the group Z/2, namely MO Z/2 * , geometric equivariant bordism n Z/2 * , and their quotient as modules over geometric bordism. This quotient is a module of stable transversality obstructions. We construct these rings from knowledge of their localizations.

Computations of complex equivariant bordism rings

We give explicit computations of the coefficients of homotopical complex equivariant cobordism theory MU G , when G is abelian. We present a set of generators which is complete for any abelian group. We present a set of relations which is complete when G is cyclic and which we conjecture to be complete in general. We proceed by first computing the localization of MU G obtained by inverting Euler classes of representations. We then define a family of operations which essentially divide by Euler classes and use these operations to define our generating sets. We give geometric applications of these computations to the study of equivariant genera, circle actions on four-manifolds, and cobordism relations between Lens spaces.

Bordisme de S 1 -fibrés vectoriels munis¶de structures supplémentaires

Bordism of S1-vector bundles with additional structures We give isomorphisms between equivariant bordism groups of certain S1-vector bundles and bordism groups of suitable “classifying” spaces determined by certain caracterestic classes. In the spinorial case, we detect the even or odd type of the S1-action and give a relationship with elleptic homology. Furthermore, we define a new type of $S^1$-actions, depending on the actions and the given slice type. This new type differs, in certain cases, from the classical odd or even type of S1-actions on spinorial manifolds.

The geometry of the local cohomology filtration in equivariant bordism

We present geometric constructions which realize the local cohomology Þltration in the setting of equivariant bordism, with the aim of understanding free G actions on manifolds. We begin by reviewing the basic construction of the local cohomology Þltration, starting with the Conner-Floyd tom Dieck exact sequence. We deÞne this Þltration geometrically using the language of families of subgroups. We then review Atiyah-SegalWilson K-theory invariants, which are well-suited for studying the manifolds produced by our techniques. We end by indicating potential applications of these ideas.

The union of a connected manifold and a point as fixed set of commuting involutions

Let A be the family of all equivariant bordism classes of involutions containing a representative ( M, T) with M connected and with the fixed point set of T being the disjoint union of a fixed connected n-dimensional manifold V n and a point. In this paper we analyse the influence of A in the determination of the equivariant bordism classes of ( Z 2) κ-actions which contain a representative fixing V n ∪ point. The results obtained are used to determine, up to bordism, all possible ( Z 2) κ-actions fixing RP(n) ∪ point with n odd.