By Robert J. Daverman

ISBN-10: 0080874436

ISBN-13: 9780080874432

ISBN-10: 0122042204

ISBN-13: 9780122042201

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A usc decomposition G of a space S is realized by a pseudo-isotopy if there exists a pseudo-isotopy yr of S to S such that YO = identity and G = [wi'(S) 1 s E S ) . By apseudo-isotopy yt of S to S we mean a homotopy y t : S -P S such that y t is a homeomorphism for each t E [0, 1) and Y I is a closed surjection. Similarly, by an isotopy wt of S to S we mean a homotopy y t : S -P S such that vt is a homeomorphism for each t E [0, 11. Throughout past history, a great deal of attention has been given to the decompositions that can be realized by pseudo-isotopies.

Shrinkable Decompositions consisting of all maps 7th-', where h represents a homeomorphism of S onto itself. Also, for n = 1,2, ... define Z,, = {F E 2 I diam F-'(x) < l / n for each x E S / G ) . The claim is that each Z,, is open and dense in Z . Openness is derived via a standard argument given by Hurewicz and Wallman [ 11: for each F E 2, the compactness of S provides a > 0 such that A c S and diam A 2 l/n imply diam F(A) > a ; then any F' E Z satisfying pc(F, F') < a/2 necessarily belongs to Z,,.

For each mapf: I S/G there exists a partition 0 = to < t I < t 2 < ... , k ) ) i s a m a p F : I - + Ssatisfying(l)F(t;) = s; and (2) both f ( [ t ; - l ,t ; ] ) and nF([t;-],t ; ] ) are contained in some V; E 1? , k. 6 . Suppose G is a monotone usc decomposition of a locally pathwise-connected Hausdorff space S and S / G has a basis of contractible (or even simply connected) open subsets. Then x: S --t S/G induces an epimorphism of fundamental groups. 7. There is no monotone map of S" onto S' x S k (n 2 2, k 2 0).

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Decompositions of Manifolds (Pure and Applied Mathematics (Academic Pr)) by Robert J. Daverman


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