By Robert J. Daverman
Decomposition idea experiences decompositions, or walls, of manifolds into easy items, frequently cell-like units. considering the fact that its inception in 1929, the topic has turn into an enormous instrument in geometric topology. the most aim of the e-book is to assist scholars drawn to geometric topology to bridge the space among entry-level graduate classes and learn on the frontier in addition to to illustrate interrelations of decomposition thought with different elements of geometric topology. With quite a few routines and difficulties, lots of them really hard, the e-book is still strongly prompt to everybody who's attracted to this topic. The ebook additionally includes an in depth bibliography and an invaluable index of keyword phrases, so it will probably additionally function a connection with a expert.
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Extra info for Decompositions of manifolds
Cellular Sets 37 Since G is shrinkable, TI is approximable by homeomorphisms. But C1 U/G is homeomorphic to R . H The topic of cellularity leads to one of the major themes of this book : the intimate connection between decomposition theory and taming theory. J. W. Cannon probably first stressed this theme, but the connections themselves have been, or should have been, visible from the outset, in the work dating back to the 1950s of R. H. Bing, E. E. Moise, and Morton Brown. Brown's important generalized Schonflies theorem [l], one of the first and perhaps the most elegant taming theorem, displays an aspect of that connection through its dependence on decomposition methods.
The map I defined as I = fn-' is light, for if C is a component of A-'(t) = nf -'(t), t E T, then z-l(C) c f -'(t) must be connected by Proposition 1, and nz-'(C) = C is necessarily a point. 4. Monotone Decompositions 19 The decomposition Mpossesses the largest possible connected elements for which fn-' is a function. The uniqueness of M follows from the observation that any decomposition having a smaller element would fail to produce a light m a p I =fn-'. A philosophical consequence of the monotone-light factorization theorem is that an understanding of all (appropriate) maps defined on a domain Scan be achieved by understanding all light maps defined on monotone images of S .
6. For each n L 2 there is a proper mapfof a connected n-manifold Monto itself having exactly one nondegenerate inverse image X and X is noncellular. 7. Let X be a compact subset of E n . Then E"/Gx is a manifold iff X is cellular. 8. Suppose Mis an n-manifold whose universal cover is En or S" and X i s a compact contractible subset of M for which M / G x is an n-manifold. Then X is cellular. 9. If the suspension CX of a compact metric space X is an n-manifold, then ZX is homeomorphic to S". 10.