708 An Alpine Bouquet of Algebraic Topology Alpine Algebraic and Applied Topology Conference August 15–21, 2016 Saas-Almagell, Switzerland Christian Ausoni Kathryn Hess Brenda Johnson Ieke Moerdijk Jérôme Scherer Editors 708 An Alpine Bouquet of Algebraic Topology Alpine Algebraic and Applied Topology Conference August 15–21, 2016 Saas-Almagell, Switzerland Christian Ausoni Kathryn Hess Brenda Johnson Ieke Moerdijk Jérôme Scherer Editors EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Catherine Yan 2010 Mathematics Subject Classification. Primary 11S40, 18Axx, 18Gxx, 19Dxx, 55Nxx, 55Pxx, 55Sxx, 55Uxx, 57Nxx. Library of Congress Cataloging-in-Publication Data Names: Conference on Alpine Algebraic and Applied Topology (2016: Saas-Almagell, Switzerland) | Ausoni, Christian, 1968– editor. Title: An alpine bouquet of algebraic topology: Conference on Alpine Algebraic and Applied Topology, August 15–21, 2016, Saas-Almagell, Switzerland / Christian Ausoni [and four oth- ers], editors. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Contem- porary mathematics; volume 708 | Includes bibliographical references. Identifiers: LCCN 2017055549 | ISBN 9781470429119 (alk. paper) Subjects: LCSH: Algebraic topology–Congresses. | AMS: Number theory – Algebraic number theory: local and p-adic fields – Zeta functions and L-functions. msc | Category theory; homological algebra – General theory of categories and functors – General theory of categories and functors. msc | Category theory; homological algebra – Homological algebra – Homological algebra. msc | K-theory – Higher algebraic K-theory – Higher algebraic K-theory. msc | Algebraic topology – Homology and cohomology theories – Homology and cohomology theories. msc | Algebraic topology – Homotopy theory – Homotopy theory. msc | Algebraic topology – Operations and obstructions – Operations and obstructions. msc | Algebraic topology – Applied homological algebra and category theory – Applied homological algebra and category theory. msc | Manifolds and cell complexes – Topological manifolds – Topological manifolds. msc Classification: LCC QA612 .C5927 2018 | DDC 514/.2–dc23 LC record available at https://lccn.loc.gov/2017055549 DOI: http://dx.doi.org/10.1090/conm/708 Color graphic policy. 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Send requests for translation rights and licensed reprints to [email protected]. c 2018 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 232221201918 Contents Preface v List of Conference Talks vii List of Participants ix Characteristics for E∞ ring spectra A. Baker 1 Segal objects and the Grothendieck construction P. Boavida de Brito 19 Blown-up intersection cohomology D. Chataur, M. Saralegi-Aranguren, and D. Tanre´ 45 Homotopically rigid Sullivan algebras and their applications C. Costoya, D. Mendez,´ and A. Viruel 103 A Dundas-Goodwillie-McCarthy theorem for split square-zero extensions of exact categories E. Dotto 123 Four approaches to cohomology theories with reality J. P. C. Greenlees 139 Topological Hochschild homology and the Hasse-Weil zeta function L. Hesselholt 157 The stable symplectic category and a conjecture of Kontsevich N. Kitchloo and J. Morava 181 Universal Gysin formulas for the universal Hall-Littlewood functions M. Nakagawa and H. Naruse 201 Graded multiplications on iterated bar constructions B. Stonek 245 Double homotopy (co)limits for relative categories K. Werndli 275 iii Preface After four conferences held in Arolla, it was time to search for new landscapes and summits. This 2016 edition of the Alpine conferences in algebraic topology took place farther east in the Saas Valley, in the village of Saas-Almagell, which offered many new challenging paths to explore. This led us to new perspectives on rational equivariant homotopy theory, topological Hochschild homology and K- theory, stable and unstable homotopy theory, functor calculus, and an incursion into applied algebraic topology. Our sincerest thanks go to all of the speakers and participants, whose talks and interest made this conference happen. We are very grateful for the authors, whose results are published here, and thank them for their contributions. Many thanks also to the referees for their anonymous work and to Chris Thivierge at the AMS for her efficient help and guidance. We would like to thank the following institutions and organizations for their financial support and sponsorship of the 2016 Alpine Algebraic and Applied Topol- ogy Conference at Saas: - U.S. National Science Foundation, - Foundation Compositio Mathematica, -Facult´e des Sciences de la Vie, Ecole´ Polytechnique F´ed´eraledeLausanne, -Conf´erence Universitaire de Suisse Occidentale, - Centre National de la Recherche Scientifique, and - Universit´e Sorbonne Paris Cit´e. This Saas edition of the Alpine topology conferences was hosted by the Hotel Kristall Saphir in Saas-Almagell. We express our gratitude to Mrs. Rosine Breuer and her family and staff for not sparing any effort in making 70 exuberant topologists feel at home. It was a pleasure organizing this conference with their help. Ch. Ausoni, K. Hess, B. Johnson, I. Moerdijk, and J. Scherer v Fluchthorn summit, August 15, 2016 List of Conference Talks Monday, August 15, 2016 Cristina Costoya Kahn’s realizability problem Tuesday, August 16, 2016 John Greenlees Rational equivariant cohomology theories and affine formal covers of the sphere David Barnes Rational S1-equivariant ring spectra Christian Wimmer Rational global homotopy theory Adeel Khan A1-homotopy invariance in spectral algebraic geometry Vesna Stojanoska Galois extensions of motivic ring spectra Nitu Kitchloo Landweber flat real pairs and ER(n)- cohomology Tse Leung So Homotopy types of gauge groups over 4-manifolds Wednesday, August 17, 2016 Vincent Franjou Lannes’ T vs Harish-Chandra restriction Anssi Lathinen Modular characteristic classes for represen- tations over finite fields Marc Hoyois On the vanishing of negative equivariant K-theory Steffen Sagave Logarithmic topological Hochschild homology Lars Hesselholt Topological Hochschild homology and the Hasse-Weil zeta function vii viii LIST OF CONFERENCE TALKS Thursday, August 18, 2016 Agn`es Beaudry The K(2)-local Picard group at p =2 Tilman Bauer Chromatic unstable homotopy, plethories, and the Dieudonn´e correspondence David Chataur Intersection homotopy theory Friday, August 19, 2016 Vidit Nanda The discrete flow category Wo jciech Chach´olski Multi-persistence Ran Levi Topological analysis of neural networks Mark Grant The Poincar´e-Hopf theorem for line fields (revisited) Wolfgang Pitsch Relative homological algebra via truncations Saturday, August 20, 2016 Pedro Boavida de Brito Configuration categories and embedding spaces Emanuele Dotto Equivariant calculus and the tower of the identity on pointed G-spaces Kristine Bauer A combinatorial model for certain Taylor towers Marc Stephan Categorical suspension and stable Postnikov data Sune Precht Reeh Saturated fusion systems as stable retracts of groups List of Participants Arlettaz, Dominique Dotto, Emanuele University of Lausanne MIT Ausoni, Christian Dundas, Bjørn Ian University Paris 13 University of Bergen Baker, Andrew Elmendorf, Tony University of Glasgow Purdue University Calumet Bandklayder, Lauren Filakovsky, Marek Northwestern University IST Austria Barnes, David Franjou, Vincent Queen’s University Belfast University of Nantes Bauer, Kristine Fresse, Benoit University of Calgary University of Lille Grant, Mark Bauer, Tilman University of Aberdeen KTH Greenlees, John Beaudry, Agn`es University of Sheffield University of Chicago Harper, John E. Boavida de Brito, Pedro Ohio State University University of Lisbon Hess, Kathryn Bobkova, Irina EPFL University of Rochester Hesselholt, Lars Chach´olski, Wojciech Nagoya University/University of KTH Copenhagen Chataur, David Hiroshi, Naruse UPJV University of Yamanashi Ching, Michael Hoyois, Marc Amherst College MIT Costoya, Cristina Jeitziner, Rachel Universidade da Coru˜na EPFL Dlotko, Pawel Joachim, Michael Inria, Saclay University of Muenster ix x LIST OF PARTICIPANTS Johnson, Brenda Sagave, Steffen Union College Radboud University Nijmegen K¸edziorek, Magdalena Scheinmann, Nathan EPFL EPFL Khan, Adeel Yusufzai Scherer, J´erˆome University Duisburg-Essen EPFL Kitchloo, Nitu