Let C denote the dual cell structure of T, let R denote the union of all
cells in C which intersect T, and let R denote the topological boundary
of R in N. A blocked space structure £ is given to R by taking the
intersections of the dual cells of C with R to be the blocks of R. The
notion of blocked space (which generalizes the notion of block bundle)
is given in [[9],section 1], The blocks of t and those of | are in a
one-one correspondence, in a way which is consistent with the boundary
operation. In fact, the 2 Z -covering of the range of t is homotopy
equivalent to R via a mapping that maps each block of this TL -covering
homotopy equivalently to the corresponding block of £. So t may be
identified with an element [t] L (£,Z ),where L (£,Z ) is a surgery
group defined in [[13], 3.3]. Roughly speaking LQ (i ,TL ) is the group
of blocked normal maps which have fundamental group TL in each block,
and are equipped with a one-one correspondence from their blocks to the
blocks of £, which is consistent with the boundary operator and shifts
dimensions down by %. The superscript "h" denotes that surgery is to
be completed only up to homotopy equivalence (not up to simple-homotopy
equivalence). The groups L„(£,Zn) are discussed in more detail in
[[13], 3.3], and similar surgery groups are described in [4].
Outline of Section 2. The blocked surgery problem t of section 1
can be studied by using the author's generalization of D. Sullivan's
Characteristic Variety Theorem (see [14]). The problem of completing
surgery on t (block by block) is thus replaced by the problem of
completing surgery on a finite set of more elementary surgery problems
t-y , tn to• The surgery problems t- are more elementary than t,
because t- has a Poincare duality space (or Z -Poincare duality space,
r=positive integer) for range and domain, and thus has at most two blocks,
where as t can have a very large number of blocks. The notion of
Z -manifold is defined in [21]; the notion of Z -Poincare duality space
is defined similarly.
In the special case that K is a PL manifold, the t- are constructed
as follows. Choose a characteristic variety for K, {g^: M. -+ K
i=l,2,3,...,£}, consisting of mappings from oriented smooth manifolds or
smooth Z -manifold (see [[14], 1.3]). Then pull the universal
Previous Page Next Page