May, J. P. Equivariant homotopy and cohomology theory. Dedicated to the memory of Robert J. Piacenza. (English) Zbl 0890.55001 Regional Conference Series in Mathematics. 91. Providence, RI: American Mathematical Society (AMS). xiii, 366 p. (1996). This is a 28 chapter volume; 13 of them are lectures presented at the NSF-CBMS Regional Conference held at the University of Alaska, Fairbanks, in August, 1993; the remaining 15 chapters introduce the reader to the main ideas and the basic tools needed to understand the lectures and were explained by the main lecturer during the conference. Chapters I and II introduce the reader to equivariant cohomology and obstruction theory, as well as localization and completion of \(G\)-spaces, where \(G\) is a compact Lie group. Chapter III, by G. Triantafillou, introduces the idea of equivariant minimal models and rational equivariant Hopf spaces, proving that they do not necessarily split as a product of Eilenberg-MacLane \(G\)-spaces in contrast with the non-equivariant case. Chapters IV and V deal with the basics of Smith theory, and with the construction of limits and colimits. The latter chapter also deals with the construction of \({\mathcal F}\)-spaces where \({\mathcal F}\) is a family of subgroups of a given group \(G\). Chapter VI, by R. Piacenza, is devoted to the homotopy, homology and cohomology theories of diagrams and, using them, provides a new proof of Elmendorf’s theorem. Chapters VII, VIII and IX give a good account of the classification of equivariant bundles; various versions of the Sullivan conjecture are given, as is also an introduction to equivariant stable homotopy theory, the latter being furnished with a quick proof of a conjecture of Conner. Chapter X, by S. Waner, deals with the theory of the so-called \(G\text{-CW}(V)\) complexes where the cells are obtained from representational spheres and some difficulties are touched upon, for example, Poincaré duality does not hold in this cellular theory. Chapter XI, by L. G. Lewis jun., contains the equivariant versions of the Hurewicz and suspension theorems. Chapters XII and XIII are good introductions to the equivariant versions of function, \(G\text{-CW}\), Eilenberg-MacLane and ring spectra as well as homology and cohomology theories graded over the real representation ring \(RO(G)\). Chapter XIV, by J. P. C. Greenlees, deals with the basics of modern equivariant \(K\)-theory, in particular with the use of the representation ring as a bundle over a point, the equivariant version of the Bott periodicity theorem and the Atiyah-Segal completion theorem. Chapter XV, by S. R. Costenoble, deals with equivariant cobordism and Thom spectra and gives some calculations in the special case of the cyclic group of order 2. Chapters XVI, XVII, XVIII, XIX and XX, explain the concepts of spectra such as fixed point and orbit spectra as well as the duality theorems of Spanier-Whitehead, Atiyah and Poincaré; prime ideals and localization of the Burnside ring are explained; we also find here transfer maps, Mackey functors and the Segal conjecture. Chapter XXI, by J. P. C. Greenlees and J. P. May, is about Tate cohomology; it explains the Tate version of Atiyah-Hirzebruch spectral sequences, cohomotopy and generalizations to families. Chapter XXII, by M. Cole, contains a survey on the homotopy properties of the twisted half-smash products and the function spectra. Chapter XXIII shows the basics of the so-called Brave New Algebra dealing with the spectra called \(S\)-algebras and \(R\)-algebras and also with topological Hochschild homology and cohomology. Chapter XXIV, by A. D. Elmendorf, L. G. Lewis jun. and J. P. May constructs the category of \(\L\)-spectra, as well as equivariant algebras and modules. Chapters XXV and XXVI, by J. P. C. Greenlees and J. P. May, gives the equivariant counterparts of the so-called Brave New Algebra stating brave new versions of local and Čech cohomology. We also find an outlined proof of the localization theorem for stable complex cobordism theory. Chapter XXVII, by G. Comenzaña and J. P. May has a rigorous proof of the completion theorem for complex cobordism using Gysin sequences. Chapter XXVIII, by G. Comenzaña, ends the volume explaining some results on the homology and cohomology theories represented by the spectrum \(MU_G\) and provides proofs of some results claimed without proof in the literature. Reviewer: J.A.Pérez (Zacatecas) Cited in 3 ReviewsCited in 107 Documents MSC: 55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 19-02 Research exposition (monographs, survey articles) pertaining to \(K\)-theory 18-02 Research exposition (monographs, survey articles) pertaining to category theory Keywords:equivariant cohomology; equivariant minimal models; equivariant Hopf spaces; Smith theory; equivariant bundles; equivariant stable homotopy theory; spectra; equivariant \(K\)-theory; equivariant cobordism; Thom spectra; Tate cohomology; brave new algebra; completion Biographic References: Piacenza, Robert J. PDFBibTeX XMLCite \textit{J. P. May}, Equivariant homotopy and cohomology theory. Dedicated to the memory of Robert J. Piacenza. Providence, RI: AMS, American Mathematical Society (1996; Zbl 0890.55001)