This book is concerned with constructing equivariant ordinary homology and cohomology for a compact Lie group \(G\). The discussion is developed with equivariant Poincaré duality as the central theme. In the non-equivariant case, Poincaré duality asserts that if \(M\) is a closed oriented \(n\)-manifold, then its fundamental class \([M]\in H_n(M)\) induces by the cap product \((-)\cap [M]\) an isomorphism \(H^k(M; \mathbb R) \cong H_{n-k}(M; \mathbb R)\). Motivated by the fact that Bredon homology is not well suited for describing this isomorphism, owing to the behavior of the cap product, the authors in [Mich. Math. J. 39, No. 2, 325–351 (1992; Zbl 0765.55003)] have tried to find an equivariant homology theory in which we can have a description of the Poincaré duality isomorphism, and have succeeded in constructing it when \(G\) is finite. So perhaps we can say that the purpose of this book is to extend further the homology obtained to one which makes it possible to provide a satisfactory description of the Poincaré duality isomorphism for all smooth compact \(G\)-manifolds \(M\) with the action of any compact Lie group \(G\).
The construction of Bredon homology is based on the use of a cell complex having cells of the form \(G/H \times D^n\). But here the authors devise a cell complex formed from cells of the form \(G \times_H D(W)\) where \(D(W)\) is the unit disc in a representation \(W\) of \(H\leq G\), which enables us to achieve the aim here. For this, first, \(M\) is assumed to be \(V\)-dimensional, which means that there is a representation \(V\) of \(G\) such that at each point \(x \in M\) the tangent space at \(x\) is isomorphic to \(V\) as a representation of the stabilizer \(G_x\) of \(x\). In addition, the notion of a dimension function for \(G\) is used which consists of a subset \(\mathscr{F}(\delta)\) of subgroups of \(G\) and an assignment to each \(H \in \mathscr{F}(\delta)\) of an \(H\)-representation \(\delta(G/H)\). On the basis of this, finally, supposing that \(\mathscr{F}(\delta)\) contains every isotropy subgroup of \(M\), the authors obtain that for each such \(\delta\) there is a cellular homology theory graded on \(RO(G)\) such that it holds that if \(M\) has a fundamental class \([M]\), then \[ (-)\cap [M] : H^\alpha_{G, \delta}(M; \bar{T}) \to H^{G, \mathscr{L}-\delta}_{V-\alpha}(M; \bar{T}) \] is an isomorphism for any virtual representation \(\alpha\) of \(G\) where \(\mathscr{L}\) denotes the tangent space at the identity of \(G\) and \(\bar{T}\) is a contravariant Mackey functor over \(M\). Unfortunately, however, this isomorphism fails to recognize \(G\)-manifolds \(M\) except the ones subordinate to a single representaion similar to the one above. The authors solve this problem by constructing a homology indexed on representations of the fundamental groupoid \(\Pi_GM\) of \(M\) which is a category having the \(G\)-maps \(G/H \to M\) as objects. The result thus obtained can be found on the last pages of the book.
By reading this book the reader will certainly be able to gain a comprehensive grasp of all the key concepts and knowledge required for understanding equivariant homology and cohomology theories for compact Lie groups and thereby hopefully be able to consider respective new perspectives on this subject.
This book consists of an introduction and three chapters with bibliography and indexes of notations and subjects. The introduction contains a short historical survey of the development of this project, in which we can find related works that provide useful background. Besides, for readers not particularly familiar with the theme here, the authors give advice on how to read this book.
Contents: \(RO(G)\)-graded ordinary homology and cohomology. Parametrized homotopy theory and fundamental groupoids. \(RO(\Pi B)\)-graded ordinary homology and cohomology.