Invitations to Geometry and Topology
The purpose of the book is to present several topics in active areas of geometry and topology. Three papers are related to topology and discrete groups. The paper by A. J. Berrick contains a review of results and conjectures about perfect groups. In particular, acyclic groups are studied. The contribution by M. R. D. Bridson relates geometry and the word problem. It includes a discussion of the Filling theorem and the Dehn function. In the paper by M. C. Crabb and A. J. B. Potter, the reader can find a description of the Fuller index in the setting of equivariant fiberwise stable homotopy theory. The other five papers are related to contemporary problems in differential geometry. M. Eastwood and J. Sawon wrote an article describing the Borel-Weil construction of finite-dimensional holomorphic representations of GL(n,C) on spaces of holomorphic sections of certain vector bundles over Pn(C). The paper by M. A. Guest is an overview over contemporary results and problems in finite-dimensional Morse theory, using Grassmannians as the main motivating examples. N. Hitchin wrote a paper on a key concept both for mathematics and theoretical physics – the Dirac operator on spin manifolds. Particular attention is devoted to the case of low dimensions; it includes a discussion of Higgs bundles and of magnetic monopoles. The contribution by S. M. Salamon on Hermitian geometry discusses several aspects of the theory of complex structures, depending on the existence of compatible Riemannian metric. It contains also a discussion of the Goldberg conjecture and the theory of connections on vector bundles related to these geometric structures. The paper by J. Seade is an expository paper on indices of vector fields and characteristic classes for singular varieties. This collection of eight articles is prepared by former students of Brian Steer, and it is dedicated to him. The list of research publications of B. F. Steer is presented in an appendix. The book contains rich and interesting material covering a broad part of geometry and topology which is explained in an accessible way. It is without any doubt an excellent book for graduate students as well as for mathematicians from other fields interested in these topics.