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| Author | : John Roe |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 106 |
| Release | : 1993 |
| Genre | : Mathematics |
| ISBN | : 0821825593 |
"July 1993, volume 104, number 497 (fourth of 6 numbers)."
| Author | : Both Professors of Maths John Roe |
| Publisher | : Oxford University Press, USA |
| Total Pages | : 106 |
| Release | : 2014-08-31 |
| Genre | : MATHEMATICS |
| ISBN | : 9781470400743 |
Coarse geometry'' is the study of metric spaces from the asymptotic point of view: two metric spaces (such as the integers and the real numbers) which look the same from a great distance'' are considered to be equivalent. This book develops a cohomology theory appropriate to coarse geometry. The theory is then used to construct higher indices'' for elliptic operators on noncompact complete Riemannian manifolds. Such an elliptic operator has an index in the $K$-theory of a certain operator algebra naturally associated to the coarse structure, and this $K$-theory then pairs with the coarse cohomology. The higher indices can be calculated in topological terms thanks to the work of Connes and Moscovici. They can also be interpreted in terms of the $K$-homology of an ideal boundary naturally associated to the coarse structure. Applications to geometry are given, and the book concludes with a discussion of the coarse analog of the Novikov conjecture.
| Author | : John Roe |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 114 |
| Release | : 1996 |
| Genre | : Mathematics |
| ISBN | : 0821804138 |
Lecture notes from the conference held Aug. 1995 in Boulder, Colo.
| Author | : Steven C. Ferry |
| Publisher | : Cambridge University Press |
| Total Pages | : 386 |
| Release | : 1995-11-23 |
| Genre | : Mathematics |
| ISBN | : 0521497965 |
These volumes are the outgrowth of a conference held at the Mathematisches Forschungsinstitut Oberwolfach (Germany) on the subject of 'Novikov Conjectures, Index Theorems and Rigidity'.
| Author | : Ulrich Bunke |
| Publisher | : Springer Nature |
| Total Pages | : 248 |
| Release | : 2020-09-03 |
| Genre | : Mathematics |
| ISBN | : 3030513351 |
Providing a new approach to assembly maps, this book develops the foundations of coarse homotopy using the language of infinity categories. It introduces the category of bornological coarse spaces and the notion of a coarse homology theory, and further constructs the universal coarse homology theory. Hybrid structures are introduced as a tool to connect large-scale with small-scale geometry, and are then employed to describe the coarse motives of bornological coarse spaces of finite asymptotic dimension. The remainder of the book is devoted to the construction of examples of coarse homology theories, including an account of the coarsification of locally finite homology theories and of coarse K-theory. Thereby it develops background material about locally finite homology theories and C*-categories. The book is intended for advanced graduate students and researchers who want to learn about the homotopy-theoretical aspects of large scale geometry via the theory of infinity categories.
| Author | : Giora Dula |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 97 |
| Release | : 1994 |
| Genre | : Mathematics |
| ISBN | : 0821825895 |
Obstruction theoretic methods are introduced into isovariant homotopy theory for a class of spaces with group actions; the latter includes all smooth actions of cyclic groups of prime power order. The central technical result is an equivalence between isovariant homotopy and specific equivariant homotopy theories for diagrams under suitable conditions. This leads to isovariant Whitehead theorems, an obstruction-theoretic approach to isovariant homotopy theory with obstructions in cohomology groups of ordinary and equivalent diagrams, and qualitative computations for rational homotopy groups of certain spaces of isovariant self maps of linear spheres. The computations show that these homotopy groups are often far more complicated than the rational homotopy groups for the corresponding spaces of equivariant self maps. Subsequent work will use these computations to construct new families of smooth actions on spheres that are topologically linear but differentiably nonlinear.
| Author | : John Patrick Campbell Greenlees |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 193 |
| Release | : 1995 |
| Genre | : Mathematics |
| ISBN | : 0821826034 |
Let [italic capital]G be a compact Lie group, [italic capitals]EG a contractible free [italic capital]G-space and let [italic capitals]E~G be the unreduced suspension of [italic capitals]EG with one of the cone points as basepoint. Let [italic]k*[over][subscript italic capital]G be a [italic capital]G-spectrum. Let [italic capital]X+ denote the disjoint union of [italic capital]X and a [italic capital]G-fixed basepoint. Define the [italic capital]G-spectra [italic]f([italic]k*[over][subscript italic capital]G) = [italic]k*[over][subscript italic capital]G [up arrowhead symbol] [italic capitals]EG+, [italic]c([italic]k*[over][subscript italic capital]G) = [italic capital]F([italic capitals]EG+,[italic]k*[over][subscript italic capital]G), and [italic]t([italic]k[subscript italic capital]G)* = [italic capital]F([italic capitals]EG+,[italic]k*[over][subscript italic capital]G) [up arrowhead symbol] [italic capitals]E~G. The last of these is the [italic capital]G-spectrum representing the generalized Tate homology and cohomology theories associated to [italic]k[subscript italic capital]G. Here [italic capital]F([italic capitals]EG+,[italic]k*[over][subscript italic capital]G) is the function space spectrum. The authors develop the properties of these theories, illustrating the manner in which they generalize the classical Tate-Swan theories.
| Author | : John Roe |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 184 |
| Release | : 2003 |
| Genre | : Mathematics |
| ISBN | : 0821833324 |
Coarse geometry is the study of spaces (particularly metric spaces) from a 'large scale' point of view, so that two spaces that look the same from a great distance are actually equivalent. This book provides a general perspective on coarse structures. It discusses results on asymptotic dimension and uniform embeddings into Hilbert space.
| Author | : Kunio Murasugi |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 118 |
| Release | : 1993 |
| Genre | : Mathematics |
| ISBN | : 0821825704 |
There are three chapters to the memoir. The first defines and develops the notion of the index of a graph. The next chapter presents the general application of the graph index to knot theory. The last section is devoted to particular examples, such as determining the braid index of alternating pretzel links. A second result shows that for an alternating knot with Alexander polynomial having leading coefficient less than 4 in absolute value, the braid index is determined by polynomial invariants.
| Author | : Wolfgang Lück |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 604 |
| Release | : 2013-03-09 |
| Genre | : Mathematics |
| ISBN | : 3662046873 |
In algebraic topology some classical invariants - such as Betti numbers and Reidemeister torsion - are defined for compact spaces and finite group actions. They can be generalized using von Neumann algebras and their traces, and applied also to non-compact spaces and infinite groups. These new L2-invariants contain very interesting and novel information and can be applied to problems arising in topology, K-Theory, differential geometry, non-commutative geometry and spectral theory. The book, written in an accessible manner, presents a comprehensive introduction to this area of research, as well as its most recent results and developments.
