Dev P. Sinha
We calculate geometric and homotopical (or stable) bordism rings associated to semi-free $S^1$ actions on complex manifolds, giving explicit generators for the geometric theory. The classification of semi-free actions with isolated fixed points up to cobordism complements similar results from symplectic geometry.
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We study certain formal group laws equipped with an action of the cyclic group of order a power of $2$. We construct $C_{2^n}$-equivariant Real oriented models of Lubin-Tate spectra $E_h$ at heights $h=2^{n-1}m$ and give explicit formulas of the $C_{2^n}$-action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory $MU_{\mathbb{R}}$, and our work examines the height of the formal group laws of the Hill-Hopkins-Ravenel norms of $MU_{\mathbb{R}}$.
In this short note, we study the bordism problem for group actions on the torus and give examples of groups acting on the torus by diffeomorphisms isotopic to the identity that cannot be extended to an action on a bounding 3-manifold. This solves a question raised in the previous work of the authors.
We give bordism-finiteness results for manifolds with semi-simple group action. Consider the class of oriented manifolds which admit a circle action with isolated fixed points such that the action extends to an $S^3$-action with fixed point. We exhibit various subclasses, characterized by an upper bound for the Euler characteristic and properties of the first Pontrjagin class, which contain only finitely many oriented bordism types in any given dimension. Also we show finiteness results for homotopy complex projective spaces and complete intersections with $S^3$-action as above.
In this paper we compute homotopical bordism rings $MU^G_*$ for abelian compact Lie groups G, giving explicit generators and relations. The key constructions are operations on equivariant bordism which should play an important role in equivariant stable homotopy theory more generally. The main technique used is localization of the theory by inverting Euler classes. Applications to homotopy theory include analysis of the completion map from $MU^G_*$ to $MU^*(BG)$. Applications to geometry include classification up to cobordism of S^1 actions on stably complex four-manifolds with precisely three…
In this work the equivariant signature of a manifold with proper action of a discrete group is defined as an invariant of equivariant bordisms. It is shown that the computation of this signature can be reduced to its computation on fixed points sets equipped with their tubular neighborhoods. It is given a description of the equivariant vector bundles with action of a discrete group $G$ for the case when the action over the base is proper quasi-free, i.e. the stationary subgroup of any point is finite. The description is given in terms of some classifying space.
We define equivariant homology theories using bordism of stratifolds with a G-action, where G is a discrete group. Stratifolds are a generalization of smooth manifolds which were introduced by Kreck. He defines homology theories using bordism of suitable stratifolds. We develop the equivariant generalization of these ideas.
Representation characterization of equivariant geometric bordism
该研究针对$G_k=(\mathbb Z_2)^k$，建立了忠实$G_k$表示作为光滑闭$G_k$流形不动点数据的刻画方法，证明其与偶性条件等价，还应用该结果得到$\dim_{\mathbb Z_2}\mathcal Z_4(G_3)=32$及对应的基元。
Homological Description of Equivariant Geometric Bordism
本文利用偶性准则与对偶原理构造有限链复形$\frak{B}_\bullet((\bZ_2)^k)$，证明特定等变无向配边群与该复形同调群同构，并通过谱序列坍缩得到配边群维数的显式公式。
William Browder in his paper "Surgery and the theory of differentiable transformation groups" developed surgery techniques to study semi-free actions of S1 on homotopy spheres, under the additional assumption that the fixed point set is a homotopy sphere. He used this surgery to show how to construct such actions. In this paper, I discussed a similar theory for semi-free actions of S3 on homotopy spheres. An open problem is raised at the end of the paper.
We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams of simplicial categories. We present a classification of bundles of categories of spectra and of the associated twisted parametrized spectra. Though twisted parametrized spectra do not have global homotopy types and therefore do not have generalized homology invariants in the usua…
Smith theory says that the fixed point of a semi-free action of a group $G$ on a contractible space is ${\bb Z}_p$-acyclic for any prime factor $p$ of $G$. Jones proved the converse of Smith theory for the case $G$ is a cyclic group acting on finite CW-complexes. We extend the theory to semi-free group action on finite CW-complexes of given homotopy type, in various settings. In particular, the converse of Smith theory holds if and only if certain $K$-theoretical obstruction vanishes. We also give some examples that show the effects of different types of the $K$-theoretical obstruction.
I give a proof of the semisimplicity of the action of the geometric frobenius on etale cohomology. This proof is based on the Weil Conjectures.
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.
We compute the topological K-theory of the group C*-algebra C*_r(G) for a group extension Z^n->G->Z/m provided that the conjugation action of Z/m on Z^n is free outside the origin.
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