In this broad introduction to topology, the author searches for topological invariants of spaces, together with techniques for their calculating. Students with knowledge of real analysis, elementary group theory, and linear algebra will quickly become familiar with a wide variety of techniques and applications involving point-set, geometric, and algebraic topology. Over 139 illustrations and more than 350 problems of various difficulties help students gain a thorough understanding of the subject.

Basic Topology (Undergraduate Texts in Mathematics)

Science & Math

Armstrong, M.A.

Mathematics

Springer

The Hauptvermutung is the conjecture that any two triangulations of a poly hedron are combinatorially equivalent. The conjecture was formulated at the turn of the century, and until its resolution was a central problem of topology. Initially, it was verified for low-dimensional polyhedra, and it might have been expected that furt her development of high-dimensional topology would lead to a verification in all dimensions. However, in 1961 Milnor constructed high-dimensional polyhedra with combinatorially inequivalent triangulations, disproving the Hauptvermutung in general. These polyhedra were not manifolds, leaving open the Hauptvermu tung for manifolds. The development of surgery theory led to the disproof of the high-dimensional manifold Hauptvermutung in the late 1960's. Unfortunately, the published record of the manifold Hauptvermutung has been incomplete, as was forcefully pointed out by Novikov in his lecture at the Browder 60th birthday conference held at Princeton in March 1994. This volume brings together the original 1967 papers of Casson and Sulli van, and the 1968/1972 'Princeton notes on the Hauptvermutung' of Armstrong, Rourke and Cooke, making this work physically accessible. These papers include several other results which have become part of the folklore but of which proofs have never been published. My own contribution is intended to serve as an intro duction to the Hauptvermutung, and also to give an account of some more recent developments in the area. In preparing the original papers for publication, only minimal changes of punctuation etc.

The Hauptvermutung Book: A Collection of Papers on the Topology of Manifolds (K-Monographs in Mathematics, 1)

Ranicki, A.A.

Casson, A.J.

Sullivan, D.P.

Rourke, C.P.

Cooke, G.E.

I'm surprised that several previous reviewers have given this book low ratings. This book is far superior to the standard introductions.As someone who has studied topology for several years now, I have found that the greatest failing of many introductory texts is the inability to give a real 'feel' for the subject. By 'feel' I mean not only familiarity with the necessary tools and ways of thought needed to progress to higher levels of understanding but also experience with the kinds of problems that plague(excite?) topologists on a daily basis.Several texts proceed in the logical progression from point set topology to algebraic topology. Munkres is among the best of this style. But the logical order is not always pedagogically best, especially in topology. To start one's topology career by spending one or more semesters on point set topology is utterly ridiculous, given that such point set subtleties are to a large degree not used to study the beginnings of geometric or algebraic topology. This is how these texts fail to give students the 'feel' for topology; the student has no idea what it is that most topologists do, and in fact will not get a good idea until much later.Armstrong tries (and succeeds for the most part) in grounding concepts in real applications, the way the tools are actually used by research mathematicians. Perhaps this is part of why it may be confusing to the novice; introducing topological groups and group actions on spaces right after the section on quotient spaces may appear a bit much, but those concepts are a big part of *why* quotient spaces are so important! Incidentally, the material on quotient spaces is the most complete I've ever seen in an introductory book; Armstrong covers cones and also gluing/attaching maps.

ARMSTRONG - Basic Topology-SPRINGER (SIE)-M. A. Armstrong-2004-EDN-1

The Hauptvermutung is the conjecture that any two triangulations of a poly­ hedron are combinatorially equivalent. The conjecture was formulated at the turn of the century, and until its resolution was a central problem of topology. Initially, it was verified for low-dimensional polyhedra, and it might have been expected that furt her development of high-dimensional topology would lead to a verification in all dimensions. However, in 1961 Milnor constructed high-dimensional polyhedra with combinatorially inequivalent triangulations, disproving the Hauptvermutung in general. These polyhedra were not manifolds, leaving open the Hauptvermu­ tung for manifolds. The development of surgery theory led to the disproof of the high-dimensional manifold Hauptvermutung in the late 1960's. Unfortunately, the published record of the manifold Hauptvermutung has been incomplete, as was forcefully pointed out by Novikov in his lecture at the Browder 60th birthday conference held at Princeton in March 1994. This volume brings together the original 1967 papers of Casson and Sulli­ van, and the 1968/1972 'Princeton notes on the Hauptvermutung' of Armstrong, Rourke and Cooke, making this work physically accessible. These papers include several other results which have become part of the folklore but of which proofs have never been published. My own contribution is intended to serve as an intro­ duction to the Hauptvermutung, and also to give an account of some more recent developments in the area. In preparing the original papers for publication, only minimal changes of punctuation etc.

The Hauptvermutung Book: A Collection of Papers on the Topology of Manifolds (K-Monographs in Mathematics)

Ranicki, A.A. J.

See the best price to sell, buy, or rent books by Armstrong, M.A.. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Books by Armstrong, M.A.

See the best price to sell, buy, or rent books by Armstrong, M.A.. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Basic Topology (Undergraduate Texts in Mathematics)

The Hauptvermutung Book: A Collection of Papers on the Topology of Manifolds (K-Monographs in Mathematics, 1)

Basic Topology (Undergraduate Texts in Mathematics)

Basic Topology (Undergraduate Texts in Mathematics)

Basic Topology (Undergraduate Texts in Mathematics)

The Hauptvermutung Book: A Collection of Papers on the Topology of Manifolds (K-Monographs in Mathematics)