10 LOWELL JONES If in addition to (a), (b), (c) above it is true that (d) dim(K') = dim(K), for all i=l,2,...,£, then the argument in section 5 is completed as follows. By (a)-(d) above, and the remarks of the last paragraph, the obstructions to com- pleting surgery on (t-,6£.) and (tt,6t-) are the same. The last obstrue- nt' m' tion vanishes, because the existence of cp' : 7L xB - + B assures that surgery can be completed on t'. So surgery can be completed on {t-9&ti)t for all i. Thus cp: TL xN - + N of 0.7 exists. If (d) is not satisfied for K c N, it will be satisfied by some product inclusion KxBu = NxBu. So there will be Z x (NxBu) - NxBu which r n can be destablized to the desired action cp:Z xN - N of 0.7. Outline of Section 6. A complete classification (.u p to concordance equivalence) of the actions cp:Z xN - » N of 0.7 is given. A generalization of 0.7 is given in which N is not required to be a manifold. Some open problems are stated. §1. Reduction to a Blocked Surgery Problem t. This section reviews how the problem stated in the introduction is reduced to a surgery problem (see [13]). First a remark on how a semifree action cp:Z xN c N can be retrieved from a free action on a "block space". Let T denote a triangulation for N which also triangulates K c N (of 0.7). Let C denote the dual cell structure of T. Let R denote the regular neighborhood for K consisting of all the cells in C which are dual to the simplices of K. Let R denote the topological boundary of R in N. The set N-R is partitioned into manifold blocks by the sets listed below. 1.1 (a) e nR, where e is a cell of C dual to a simplex of K. (b) ITR, "SN-TL The set N-R together with the partitioning sets listed in 1.1 is denoted by £. The sets in 1.1 (b) will be denoted by b,(£), b ? ^ respectively. The sets in 1.1 (a) will be denoted by b.(£), 3jx, where x is the number of blocks in 1.1. In the terminology of [ [9] , §1] £ is a block space, and the b.(£) are the blocks of £. The manifold boundaries ab.CO
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