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Springer INdAM Series 14 Filippo Callegaro · Frederick Cohen Corrado De Concini · Eva Maria Feichtner Giovanni Gai · Mario Salvetti Editors Con guration Spaces Geometry, Topology and Representation Theory Springer INdAM Series Vo lu me 1 4 Editor-in-Chief G. Patrizio Series Editors C. Canuto G. Coletti G. Gentili A. Malchiodi P. Marcellini E. Mezzetti G. Moscariello T. Ruggeri More information about this volume at http://www.springer.com/series/10283 Filippo Callegaro • Frederick Cohen • Corrado De Concini • Eva Maria Feichtner • Giovanni Gaiffi • Mario Salvetti Editors Configuration Spaces Geometry, Topology and Representation Theory 123 Editors Filippo Callegaro Frederick Cohen Dipartimento di Matematica Department of Mathematics UniversitadiPisaJ University of Rochester Pisa, Italy Rochester, USA Corrado De Concini Eva Maria Feichtner Dipartimento di Matematica Department of Mathematics UniversitadiRomaLaSapienzaJ University of Bremen Roma, Italy Bremen, Germany Giovanni Gaiffi Mario Salvetti Dipartimento di Matematica Dipartimento di Matematica UniversitadiPisaJ UniversitadiPisaJ Pisa, Italy Pisa, Italy ISSN 2281-518X ISSN 2281-5198 (electronic) Springer INdAM Series ISBN 978-3-319-31579-9 ISBN 978-3-319-31580-5 (eBook) DOI 10.1007/978-3-319-31580-5 Library of Congress Control Number: 2016943640 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland Preface Configuration spaces: In by now classical notation following the seminal paper of Fadell and Neuwirth in the early 1960s, they read as n Fn.X/ Df.x1;:::;xn/ 2 X j xi ¤ xj for i ¤ jg; where X is a topological space and n a natural number. Beautifully simple at the outset and studied for decades, configuration spaces continue to thrive at the crossroads of algebra, geometry, and topology, and even attract the interest of applied fields such as phylogenetics, robotics, and distributed computing. The present volume is a result of the INdAM Conference “Configuration Spaces: Geometry, Topology, and Representation Theory”, which took place at the Palazzone in Cortona, Italy, from 31 August to 6 September 2014. This conference was an initiative of the Istituto Nazionale di Alta Matematica (INdAM) and of the Futuro in Ricerca project “Perspectives in Lie Theory”, a satellite event of the INdAM research program “Perspectives in Lie Theory” held in Pisa from December 2014 to February 2015. This volume reflects much of the spirit of the INdAM Conference. It provides a snapshot of the current developments in configuration spaces and a glimpse on what the near future holds in store. Conference participants and readers alike have the opportunity to explore the landscape of configuration spaces, knowledgeably guided by experts in the field. Let us draw a road map of this volume: Arrangement complements have for long been understood as special, though most instructive cases of configuration spaces. Rooted in the world of classical configuration spaces with a prototype example, the braid arrangement, a prominent part of this volume is devoted to arrangement theory. The volume begins with a series of chapters “Around the Tangent Cone Theorem”, “Higher Resonance Varieties of Matroids”, “Local Asymptotic Euler-Maclaurin Expansion for Rie- mann Sums Over a Semi-Rational Polyhedron”, “Leray Spectral Sequence for Complements of Certain Arrangements of Smooth Submanifolds”, and “Higher Topological Complexity of Artin Type Groups” that cover arrangement theory v vi Preface from its purest parts (resonance and characteristic varieties) over the integral geometry of polyhedra to the application-inspired realm of (higher) topological complexity. From there the volume ventures into actual applications, both within other parts of mathematics, in fact, in probabilistic topology (chapter “Random Simplicial Complexes”), and beyond, notably in theoretical distributed computing (chapter “Combinatorial Topology of the Standard Chromatic Subdivision and Weak Symmetry Breaking for 6 Processes”). There have been many instances over time where combinatorial data have been crucial for a better understanding of configuration spaces and their invariants. Matroid theory has for long been the right degree of abstraction to deal with arrangements, arrangement compactifications and the like. More recently, the classical concept of nets appeared, which is believed, if not proven, to be crucial for the understanding of Milnor fiber cohomology. Chapters “Cryptomorphisms for Abstract Rigidity Matroids” and “Induced and Complete Multinets” are of mostly combinatorial nature and pay tribute to matroids and nets as key concepts in arrangement theory. In chapters “Cohomology of the Milnor Fibre of a Hyperplane Arrangement with Symmetry” and “On the Twisted Cohomology of Affine Line Arrangements”, which address the topology of Milnor fibers of arrangement complements, we turn back to the heart of arrangement theory. Geometric group theory is considered in chapter “Braid Groups and Euclidean Simplices”, and the volume closes with three chapters providing the link to representation theory. We are very grateful to the participants of the Cortona conference for creating a unique atmosphere of interaction, and we are indebted to the authors of this volume for capturing this spirit. May this volume inspire further thought to the benefit of a multifaceted field. Finally, we wish to thank the Palazzone and INdAM staff for their very efficient job in keeping the organizational matters of the conference afloat. Pisa, Italy Filippo Callegaro Rochester, NY, USA Fred Cohen Roma, Italy Corrado De Concini Bremen, Germany Eva Maria Feichtner Pisa, Italy Giovanni Gaiffi Pisa, Italy Mario Salvetti September 2015 Contents Around the Tangent Cone Theorem ........................................... 1 Alexander I. Suciu Higher Resonance Varieties of Matroids ...................................... 41 Graham Denham Local Asymptotic Euler-Maclaurin Expansion for Riemann Sums over a Semi-Rational Polyhedron ....................................... 67 Nicole Berline and Michèle Vergne Leray Spectral Sequence for Complements of Certain Arrangements of Smooth Submanifolds ....................................... 107 Andrzej Weber Higher Topological Complexity of Artin Type Groups ...................... 119 Sergey Yuzvinsky Random Simplicial Complexes ................................................. 129 Armindo Costa and Michael Farber Combinatorial Topology of the Standard Chromatic Subdivision and Weak Symmetry Breaking for Six Processes .............. 155 Dmitry N. Kozlov Cryptomorphisms for Abstract Rigidity Matroids ........................... 195 Emanuele Delucchi and Tim Lindemann Induced and Complete Multinets .............................................. 213 Jeremiah Bartz Cohomology of the Milnor Fibre of a Hyperplane Arrangement with Symmetry ................................................................... 233 Alexandru Dimca and Gus Lehrer On the Twisted Cohomology of Affine Line Arrangements ................. 275 Mario Salvetti and Matteo Serventi vii viii Contents Braid Groups and Euclidean Simplices........................................ 291 Elizabeth Leyton Chisholm and Jon McCammond The Cohomology of M0;n as an FI-Module .................................... 313 Rita Jiménez Rolland Syzygies in Equivariant Cohomology for Non-abelian Lie Groups......... 325 Matthias Franz A Survey on Spaces of Homomorphisms to Lie Groups ..................... 361 Frederick R. Cohen and Mentor Stafa Contributors Jeremiah Bartz Francis Marion University, Florence, SC, USA Nicole Berline École Polytechnique, Centre de Mathématiques Laurent Schwartz, Palaiseau Cedex, France Frederick R. Cohen University of Rochester, Rochester, NY, USA Armindo Costa School of Mathematical Sciences, Queen Mary University of London, London, UK Emanuele Delucchi Department of Mathematics, University of Fribourg, Fri- bourg, Switzerland Graham Denham Department of Mathematics, University of Western Ontario, London, Canada Alexandru Dimca University of Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, Nice, France Michael Farber School of Mathematical Sciences, Queen Mary University of London, London, UK Matthias Franz Department of Mathematics, University of Western Ontario, London, Canada Rita Jiménez Rolland Centro de Ciencias Matemáticas, Universidad Nacional