4 LOWELL JONES

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