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Equivariant Homotopy and Cohomology EQUIVARIANT HOMOTOPY AND COHOMOLOGY THEORY May JP The author acknowledges the supp ort of the NSF Mathematics Subject Classication Revision Primary L M N N N N N P P P P Q R R T R Secondary E G G A P P P Q Q R R S S S U U R R S Author addresses University of Chicago Chicago Il Email address maymathuchicagoedu ii Contents Intro duction Chapter I Equivariant Cellular and Homology Theory Some basic denitions and adjunctions Analogs for based Gspaces GCW complexes Ordinary homology and cohomology theories Obstruction theory ersal co ecient sp ectral sequences Univ Chapter I I P ostnikov Systems Lo calization and Completion Gspaces and Postnikov systems Eilenb ergMacLane lo calizations of spaces Summary Lo calizations of Gspaces Summary completions of spaces Completions of Gspaces Chapter I I I Equivariant Rational Homotopy Theory Summary the theory of minimal mo dels Equivariant minimal mo dels Rational equivariant Hopf spaces Chapter IV Smith Theory cohomology Smith theory via Bredon iii iv CONTENTS Borel cohomology lo calization and Smith theory Equivariant Applications Chapter V Categorical Constructions Co ends and geometric realization Homotopy colimits and limits Elmendorf s theorem on diagrams of xed p oint spaces spaces Eilenb ergMac Lane Gspaces and universal F Chapter VI The Homotopy Theory of Diagrams Elementary homotopy theory of diagrams Homotopy Groups Cellular Theory The homology and cohomology theory of diagrams J The closed mo del structure on U Another pro of of Elmendorf s theorem Chapter VI I Equivariant Bundle theory and Classifying Spaces The denition of equivariant bundles The classication of equivariant bundles Some examples of classifying spaces Chapter VI I I The Sullivan Conjecture Statements of versions of the Sullivan conjecture and Fix Algebraic preliminaries Lannes functors T Lannes generalization of the Sullivan conjecture Sketch pro of of Lannes theorem Maps b etw een classifying spaces An intro duction to equivariant stable homotopy Chapter IX Gspheres in homotopy theory GUniverses and stable Gmaps Euler characteristic and transfer Gmaps Mackey functors and coMackey functors Ggraded homology and cohomology RO CONTENTS v The Conner conjecture V complexes and RO Ggraded cohomology Chapter X GCW Motivation for cellular theories based on representations GCWV complexes Homotopy theory of GCWV complexes Ordinary RO Ggraded homology and cohomology Chapter XI The equivariant Hurewicz and Susp ension Theorems Background on the classical theorems Formulation of the problem and counterexamples An oversimplied description of the results The statements of the theorems Sketch pro ofs of the theorems Chapter XI I The Equivariant Stable Homotopy Category An intro ductory overview Presp ectra and sp ectra Smash pro ducts Function sp ectra The equivariant case Spheres and homotopy groups GCW sp ectra Stability of the stable category Getting into the stable category Chapter XI I I RO Ggraded homology and cohomology theories Axioms for ROGgraded cohomology theories Representing RO Ggraded theories by Gsp ectra Browns theorem and RO Ggraded cohomology Equivariant Eilenb ergMacLane sp ectra Ring Gsp ectra and pro ducts theory Chapter XIV An intro duction to equivariant K vi CONTENTS The denition and basic prop erties of K theory G Bundles over a p oint the representation ring Equivariant Bott p erio dicity Equivariant K theory sp ectra The AtiyahSegal completion theorem The generalization to families Chapter XV An intro duction to equivariant cob ordism A review of nonequivariant cob ordism Equivariant cob ordism and Thom sp ectra Computations the use of families Sp ecial cases o dd order groups and Z Chapter XVI Sp ectra and Gsp ectra change of groups duality Fixed p oint sp ectra and orbit sp ectra Split Gsp ectra and free Gsp ectra Geometric xed p oint sp ectra uller isomorphism Change of groups and the Wirthm Quotien t groups and the Adams isomorphism sp ectra from Gsp ectra The construction of GN SpanierWhitehead duality V duality of Gspaces and Atiyah duality Poincar e duality Chapter XVI I The Burnside ring Generalized Euler characteristics and transfer maps G The Burnside ring AG and the zero stem S 0 Prime ideals of the Burnside ring Idemp otent elements of the Burnside ring Lo calizations of the Burnside ring Lo calization of equivariant homology and cohomology Chapter XVI I I Transfer maps in equivariant bundle theory CONTENTS vii The transfer and a dimensionshifting variant Basic prop erties of transfer maps Smash pro ducts and Euler characteristics The double coset formula and its applications Transitivity of the transfer Chapter XIX Stable homotopy and Mackey functors The splitting of equivariant stable homotopy groups Generalizations of the splitting theorems Equivalent denitions of Mackey functors Induction theorems Splittings of rational Gsp ectra for nite groups G Chapter XX The Segal conjecture The statement in terms of completions of Gsp ectra A calculational reformulation groups A generalization and the reduction to p The pro of of the Segal conjecture for nite pgroups Approximations of singular subspaces of Gspaces An inverse limit of Adams sp ectral sequences Further generalizations maps b etween classifying spaces Chapter XXI Generalized Tate cohomology Denitions and basic prop erties Ordinary theories AtiyahHirzebruch sp ectral sequences Cohomotopy p erio dicity and ro ot invariants The generalization to families Equivariant K theory Further calculations and applications Chapter XXI I Brave new algebra The category of S mo dules Categories of Rmo dules viii CONTENTS The algebraic theory of Rmo dules The homotopical theory of Rmo dules Categories of Ralgebras Bouseld lo calizations of Rmo dules and algebras Top ological Ho chschild homology and cohomology Chapter XXI I I Brave new equivariant foundations Twisted halfsmash pro ducts sp ectra The category of L A ring sp ectra and S algebras and E 1 1 Alternative p ersp ectives on equivariance The construction of equivariant algebras and mo dules Comparisons of categories of LGsp ectra Equivariant Algebra Chapter XXIV Brave New Intro duction Cech cohomology in algebra Lo cal and Brave new versions of lo cal and Cech cohomology Lo calization theorems in equivariant homology Completions completion theorems and lo cal homology A pro of and generalization of the lo calization theorem The application to K theory Lo cal Tate cohomology Chapter XXV Lo calization and completion in complex b ordism The lo calization theorem for stable complex b ordism An outline of the pro of The norm map and its prop erties The idea b ehind the construction of norm maps functors with smash pro duct Global I The denition of the norm map The splitting of M U as an algebra G CONTENTS ix Loers completion conjecture Chapter XXVI Some calculations in complex equivariant b ordism Notations and terminology Stably almost complex structures and b ordism Tangential structures Calculational to ols Statements of the main results 1 S Preliminary lemmas and families in G 1 On the families F in G S i 1 Passing from G to G S and G Z k Bibliography Intro duction This volume b egan with Bob Piacenzas suggestion that I b e the principal lecturer Regional Conference in Fairbanks Alaska That event to ok at an NSFCBMS place in August of and the interim has seen very substantial progress in this general area of mathematics The scop e of this volume has grown accordingly ariant algebraic top ology to sta The original fo cus was an intro duction to equiv ble homotopy theory and to equivariant stable homotopy theory that was geared towards graduate students with a reasonably go o d understanding of nonequivari ant algebraic top ology More recent material is changing the direction of the last by allowing the intro duction of p ointset top ological algebra into sta two sub jects ble homotopy theory b oth equivariant and nonequivariant and the last p ortion of the b o ok fo cuses on an intro duction to these new developments There is a progression with the later p ortions of the b o ok on the whole b eing more dicult than the earlier p ortions Equivariant algebraic top ology concerns the study of algebraic invariants of two chapters intro duce the basic structural spaces with group actions The rst foundations of the sub ject cellular theory ordinary homology and cohomology theory Eilenb ergMac Lane Gspaces Postnikov systems lo calizations of Gspaces and completions of Gspaces In most of this work G can b e any top ological group but we restrict attention to compact Lie groups in the rest of the b o ok Chapter I I I on equivariant rational homotopy theory was written by Georgia Triantallou In it she shows how to generalize Sullivans theory of minimal mo dels to obtain an algebraization of the homotopy category of nilp otent G spaces for a nite group G This chapter contains a rst surprise rational Hopf Gspaces need not split as pro ducts of Eilenb ergMac Lane Gspaces This is a hint INTRODUCTION that the calculational b ehavior of equivariant algebraic top ology is more intricate and dicult to determine than that of the classical nonequivariant theory gives two pro ofs of the rst main theorem of equivariant algebraic Chapter IV theory any xed p oint space
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