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Open Problems in the Topology of Manifolds

Abstract

Not specified

Authors: Jonathan Bowden, Diarmuid Crowley, Jim Davis, Stefan Friedl, Carmen Rovi, Stephan Tillmann

Date Published: 2021

Publication Type: InBook

Signature cocycles on the mapping class group and symplectic groups

Abstract (Expand)

Werner Meyer constructed a cocycle in H^2(Sp_2(g,Z); Z) which computes the signature of a closed oriented surface bundle over a surface. By studying properties of this cocycle, he also showed that the … signature of such a surface bundle is a multiple of 4. In this paper, we study signature cocycles both from the geometric and algebraic points of view. We present geometric constructions which are relevant to the signature cocycle and provide an alternative to Meyer's decomposition of a surface bundle. Furthermore, we discuss the precise relation between the Meyer and Wall-Maslov index. The main theorem of the paper, Theorem 6.6, provides the necessary group cohomology results to analyze the signature of a surface bundle modulo any integer N. Using these results, we are able to give a complete answer for N=2,4 and 8, and based on a theorem of Deligne, we show that this is the best we can hope for using this method.

Authors: Dave Benson, Caterina Campagnolo, Andrew Ranicki, Carmen Rovi

Date Published: 1st Nov 2020

Publication Type: Journal

Flows on the $$\mathbf{PGL(V)}$$-Hitchin Component

Abstract (Expand)

In this article we define new flows on the Hitchin components for PSL(n,R). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other … examples, so called eruption flows, are associated to pair of pants in S and capture new phenomena which are not present in the case when n = 2. We determine a global coordinate system on the Hitchin component. Using the computation of the Goldman symplectic form on the Hitchin component, that is developed by two of the authors in a companion paper to this article (Sun and Zhang in The Goldman symplectic form on the PGL(V)-Hitchin component, 2017. arXiv:1709.03589), this gives a global Darboux coordinate system on the Hitchin component.

Authors: Zhe Sun, Anna Wienhard, Tengren Zhang

Date Published: 1st Apr 2020

Publication Type: Journal

Topological mixing of Weyl chamber flows

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In this paper we study topological properties of the right action by translation of the Weyl chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological … mixing.

Authors: NGUYEN-THI DANG, OLIVIER GLORIEUX

Date Published: 17th Feb 2020

Publication Type: Journal

Renormalizing an infinite rational IET

Abstract (Expand)

We study an interval exchange transformation of [0,1] formed by cutting the interval at the points 1/n and reversing the order of the intervals. We find that the transformation is periodic away from a … Cantor set of Hausdorff dimension zero. On the Cantor set, the dynamics are nearly conjugate to the 2-adic odometer.

Authors: W. Patrick Hooper, Kasra Rafi, Anja Randecker

Date Published: 2020

Publication Type: Journal

The real spectrum compactification of character varieties: characterizations and applications

Abstract (Expand)

We announce results on a compactification of general character varieties that has good topological properties and give various interpretations of its ideal points. We relate this to the Weyl chamber … length compactification and apply our results to the theory of maximal and Hitchin representations.

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Date Published: No date defined

Publication Type: Unpublished

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