Intersection Spaces, Spatial Homology Truncation, and String Theory
Intersection cohomology assigns groups which satisfy a generalized form of Poincaré duality over the rationals to a stratified singular space. The present monograph introduces a method that assigns to certain classes of stratified spaces cell complexes, c
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1997
Markus Banagl
Intersection Spaces, Spatial Homology Truncation, and String Theory
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Markus Banagl University of Heidelberg Mathematics Institute Im Neuenheimer Feld 288 69120 Heidelberg Germany [email protected]
ISBN: 978-3-642-12588-1 e-ISBN: 978-3-642-12589-8 DOI: 10.1007/978-3-642-12589-8 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2010928327 Mathematics Subject Classification (2000): 55N33; 57P10; 14J17; 81T30; 55P30; 55S36; 14J32; 14J33 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
Preface
The primary concern of the work presented here is Poincar´e duality for spaces that are not manifolds, but are still put together from manifolds that form the strata of a stratification of the space. Goresky and MacPherson’s intersection homology [GM80, GM83], see also [B+ 84, KW06, Ban07], associates to a stratified pseudo¯ manifold X chain complexes IC∗p¯ (X; Q) depending on a perversity parameter p, whose homology IH∗p¯ (X; Q) = H∗ (IC∗p¯ (X; Q)) satisfies generalized Poincar´e duality across complementary perversities when X is closed and oriented. L2 -cohomology [Che80, Che79, Che83] associates to a triangulated pseudomanifold X equipped with a suitable conical Riemannian metric on the top stratum a differential com∗ (X), the complex of differential L2 -forms ω on the top stratum of X such plex Ω(2) ∗ (X) = H ∗ (Ω ∗ (X)) satisfies Poincar´ e that d ω is L2 as well, whose cohomology H(2) (2) duality (at least when X has no strata of odd codimension; in more general situations one must choose certain boundary conditions). The linear dual of IH∗m¯ (X; R) ∗ (X), by integration. In the present work, we adopt the “spatial is isomorphic to H(2) philosophy” outlined in the announcement [Ban09], maintaining that a theory of Poincar´e duality for stratified spaces benefits from being implemented on the level of spaces, with passage to coarser filters such as chain complexes, homology or homotopy groups occurring as late as possible in the course of the developmen
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