Exotic Smoothness and Physics: Differential Topology and by Torsten Asselmeyer-Maluga

By Torsten Asselmeyer-Maluga

The new revolution in differential topology with regards to the invention of non-standard ("exotic") smoothness constructions on topologically trivial manifolds similar to R4 indicates many interesting possibilities for purposes of probably deep value for the spacetime types of theoretical physics, particularly common relativity. This wealthy panoply of recent differentiable buildings lies within the formerly unexplored zone among topology and geometry. simply as actual geometry used to be considered trivial earlier than Einstein, physicists have endured to paintings lower than the tacit - yet now proven to be flawed - assumption that differentiability is uniquely made up our minds by means of topology for easy four-manifolds. seeing that diffeomorphisms are the mathematical versions for actual coordinate differences, Einstein's relativity precept calls for that those versions be bodily inequivalent. This e-book offers an introductory survey of a few of the suitable arithmetic and provides initial effects and proposals for additional functions to spacetime versions.

Show description

Read Online or Download Exotic Smoothness and Physics: Differential Topology and Spacetime Models PDF

Best geometry and topology books

Differential Geometry. Proc. conf. Lyngby, 1985

The Nordic summer season college 1985 awarded to younger researchers the mathematical elements of the continuing study stemming from the examine of box theories in physics and the differential geometry of fibre bundles in arithmetic. the amount comprises papers, usually with unique strains of assault, on twistor tools for harmonic maps, the differential geometric elements of Yang-Mills concept, complicated differential geometry, metric differential geometry and partial differential equations in differential geometry.

Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 1986–87

This is often the 3rd released quantity of the court cases of the Israel Seminar on Geometric facets of sensible research. the big majority of the papers during this quantity are unique examine papers. there has been final 12 months a powerful emphasis on classical finite-dimensional convexity thought and its reference to Banach area thought.

Lectures on the geometry of quantization

Those notes are in keeping with a path entitled "Symplectic Geometry and Geometric Quantization" taught by means of Alan Weinstein on the college of California, Berkeley (fall 1992) and on the Centre Emile Borel (spring 1994). the one prerequisite for the path wanted is a data of the elemental notions from the idea of differentiable manifolds (differential varieties, vector fields, transversality, and so forth.

Extra info for Exotic Smoothness and Physics: Differential Topology and Spacetime Models

Sample text

Let A c X, and f : A -+ Y, then define Y f = X Uf Y as the space obtained from X U Y by identifying z E A c X with f(x) E Y. An important special case is the attachment of cells, X = D n , A = dDn = - a sn-1 This leads to the definition of an important special class of spaces, the CW complexes as spaces defined by a decomposition, xoc x1 f a - c x n = x, in which X' is a finite set of points and Xk is obtained from Xk-l by attaching a finite number of k-cells. 6 Axiomatic Homology Theory In addition to the singular approach to homology there is also the simplicia1 one we mentioned briefly in the introduction, and at least two other fairly Exotic Smoothness and Physics 34 well-known approaches for cohomology, those of Cech and dertham.

Since these structures depend only on the topology, we can say that two spaces with different algebraic structures are not homeomorphic. However, in general the converse will not be true. One of the most important long standing problems in mathematics revolves around this issue for the apparently elementary space, S 3 . The Poincar6 conjecture, described more carefully below, is that any topological manifold with the same homotopy groups Algebraic Tools f o r Topology 21 as standard S3 must necessarily be homeomorphic to it.

The reciprocal relationship of cancellation between products and quotients familiar in ordinary arithmetic does not hold in the topological category. A very interesting counter example is provided by Whitehead manifolds, first defined in 1935, [Whitehead (1935)l. These are open, contractible 3-manifolds, W, that are not homeomorphic t o standard W3, but, as shown by Glimm, [Glimm (1960)], the topological product W’ x W is homeomorphic to W4. Thus, in this product, “factoring out” W’ from W4 does not necessarily result in w3.

Download PDF sample

Rated 4.07 of 5 – based on 16 votes