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