ISSN numbers are printed here 1 Geometry & Topology Monographs Volume X: Volume name goes here Pages 1{XXX The obstruction to ¯nding a boundary for an open manifold of dimension greater than ¯ve Laurence C. Siebenmann Abstract For dimensions greater than ¯ve the main theorem gives nec- essary and su±cient conditions that a smooth open manifold W be the interior of a smooth compact manifold with boundary. The basic necessary condition is that each end ² of W be tame. Tameness consists of two parts (a) and (b): (a) The system of fundamental groups of connected open neighborhoods of ² is stable. This means that (with any base points and connecting paths) there exists a co¯nal sequence o f1 o f2 G1 G2 ::: so that isomorphisms are induced »= »= Image(f1)o Image(f2) o ::: : (b) There exist arbitrarily small open neighborhoods of ² that are dominated each by a ¯nite complex. Tameness for ² clearly depends only on the topology of W . It is shown that if W is connected and of dimension > 5, its ends are all tame if and only if W £S1 is the interior of a smooth compact manifold. However examples of smooth open manifolds W are constructed in each dimension > 5 so that W itself is not the interior of a smooth compact manifold although W £ S1 is. When (a) holds for ², the projective class group Kf (¼ (²)) = lim G 0 1 á j j is well de¯ned up to canonical isomorphism. When ² is tame an invariant σ(²) 2 Kf0(¼1²) is de¯ned using the smoothness structure as well as the topology of W . It is closely related to Wall's obstruction to ¯niteness for CW complexes (Annals of Math. 81 (1965) pp. 56-69). Copyright declaration is printed here 2 Laurence C. Siebenmann Main Theorem. A smooth open manifold W n , n > 5, is the interior of a smooth compact manifold if and only if W has ¯nitely many connected components, and each end ² of W is tame with invariant σ(²) = 0. (This generalizes a theorem of Browder, Levine, and Livesay, A.M.S. Notices 12, Jan. 1965, 619-205). For the study of σ(²), a sum theorem and a product theorem are established for C.T.C. Wall's related obstruction. Analysis of the di®erent ways to ¯t a boundary onto W shows that there exist smooth contractible open subsets W of Rn , n odd, n > 5, and di®eomorphisms of W onto itself that are smoothly pseudo-isotopic but not smoothly isotopic. The main theorem can be relativized. A useful consequence is Proposition. Suppose W is a smooth open manifold of dimension > 5 and N is a smoothly and properly imbedded submanifold of codimension k 6= 2. Suppose that W and N separately admit completions. If k = 1 suppose N is 1-connected at each end. Then there exists a compact manifold pair (W; N) such that W = Int W , N = Int N . If W n is a smooth open manifold homeomorphic to M £ (0; 1) where M is a closed connected topological (n ¡ 1)-manifold, then W has two ends ²¡ and ²+ , both tame. With ¼1(²¡) and ¼1(²+) identi¯ed with ¼1(W ) n¡1 there is a duality σ(²+) = (¡1) σ(²¡) where the bar denotes a certain involution of the projective class group Kf0(¼1W ) analogous to one de¯ned by J.W. Milnor for Whitehead groups. Here are two corollaries. If M m is a stably smoothable closed topological manifold, the obstruction σ(M) to M having the homotopy type of a ¯nite complex has the symmetry σ(M) = (¡1)mσ(M). If ² is a tame end of an open topological manifold n W and ²1 , ²2 are the corresponding smooth ends for two smoothings of W , n then the di®erence σ(²1) ¡ σ(²2) = σ0 satis¯es σ0 = (¡1) σ0 . Warning: In case every compact topological manifold has the homotopy type of a ¯nite complex all three duality statements above are 0 = 0. It is widely believed that all the handlebody techniques used in this thesis have counterparts for piecewise-linear manifolds. Granting this, all the above results can be restated for piecewise-linear manifolds with one slight exception. For the proposition on pairs (W; N) one must insist that N be locally unknotted in W in case it has codimension one. AMS Classi¯cation Keywords October 27, 2007 Geometry & Topology Monographs, Volume X (20XX) The obstruction to ¯nding a boundary for an open manifold 3 Introduction The starting point for this thesis is a problem broached by W. Browder, J. Levine, and G. R. Livesay in [1]. They characterize those smooth open mani- folds W w , w > 5 that form the interior of some smooth compact manifold W with a simply connected boundary. Of course, manifolds are to be Hausdor® and paracompact. Beyond this, the conditions are (A) There exists arbitrarily large compact sets in W with 1{connected com- plement. (B) H¤(W ) is ¯nitely generated as an abelian group. I extend this characterization and give conditions that W be the interior of any smooth compact manifold. For the purposes of this introduction let W w be a connected smooth open manifold, that has one end { i.e. such that the complement of any compact set has exactly one unbounded component. This end { call it " { may be identi¯ed with the collection of neighborhoods of 1 of W . " is said to be tame if it satis¯es two conditions analogous to (A) and (B): (a) ¼1 is stable at ". (b) There exists arbitrarily small neighborhoods of ", each dominated by a ¯nite complex. When " is tame an invariant σ(") is de¯ned, and for this de¯nition, no re- striction on the dimension w of W is required. The main theorem states that if w > 5, the necessary and su±cient conditions that W be the interior of a smooth compact manifold are that " be tame and the invariant σ(") be zero. Examples are constructed in each dimension > 5 where " is tame but σ(") 6= 0. For dimensions > 5, " is tame if and only if W £ S1 is the interior of a smooth compact manifold. The stability of ¼1 at " can be tested by examining the fundamental group system for any convenient sequence Y1 ⊃ Y2 ⊃ ::: of open connected neighbor- hoods of " with \ closure(Yi) = ;: i If ¼ is stable at ", ¼ (") = lim ¼ (Y ) is well de¯ned up to isomorphism in a 1 1 á 1 i preferred conjugacy class. Condition (b) can be tested as follows. Let V be any closed connected neighbor- hood of " which is a topological manifold (with boundary) and is small enough so that ¼1(") is a retract of ¼1(V ) { i.e. so that the natural homomorphism Geometry & Topology Monographs, Volume X (20XX) 4 Laurence C. Siebenmann ¼1(") ¡! ¼1(V ) has a left inverse. (Stability of ¼1 at " guarantees that such a neighborhood exists.) It turns out that condition (b) holds if and only if V is dominated by a ¯nite complex. No condition on the homotopy type of W can replace (b), for there exist contractible W such that (a) holds and ¼1(") is even ¯nitely presented, but " is, in spite of this, not tame. On the other hand, tameness clearly depends only on the topology of W . The invariant σ(") of a tame end " is an element of the group Kf0(¼1") of stable isomorphism classes of ¯nitely generated projective modules over ¼1("). If, in testing (b) one chooses the neighborhood V of " (above) to be a smooth submanifold, then σ(") = r¤σ(V ) where σ(V ) 2 Kf0(¼1V ) is up to sign C.T.C. Wall's obstruction [2] to V having the homotopy type of a ¯nite complex, and r¤ is induced by a retraction of ¼1(V ) onto ¼1("). Note that σ(") seems to depend on the smoothness struc- ture of W . For example, every tame end of dimension at least 5 has arbitrarily small open neighborhoods each homotopy equivalent to a ¯nite complex (use Theorem 8.6 and Theorem 6.5). The discussion of tameness and the de¯nition for σ(") is scattered in various sections. The main references are: De¯nitions 3.6, 4.2, Proposition 4.3, De¯nition 4.4, Lemma 6.1, De¯nition 7.7, x10, Corol- lary 11.6. The proof of the main theorem applies the theory of non-simply connected handles bodies as expounded in Barden [31] and Wall [3] to ¯nd a collar for ", viz. a closed neighborhood V which is a smooth submanifold di®eomorphic with BdV £ [0; 1). In dimension 5, the proof breaks down only because Whitney's famous device fails to untangle 2{spheres in 4{manifolds (c.f. x5). In dimension 2, tameness alone ensures that a collar exists (see Kerekjarto [26, p. 171]). It seems possible that the same is true in dimension 3 (modulo the Poincar¶e Conjecture) { c.f. Wall [30]. Dimension 4 is a complete mystery. There is a striking parallelism between the theory of tame ends developed here and the well known theory of h{cobordisms. For example the main theorem corresponds to the s{cobordism theorem of B. Mazur [34][3]. The relationship can be explained thus. For a tame end " of dimension > 6 the invariant σ(") 2 Kf0(¼1") is the obstruction to ¯nding a collar. When a collar exists, parallel families of collars are classi¯ed relative to a ¯xed collar by torsions ¿ 2 Wh(¼1") of certain h{cobordisms (c.f. Theorem 9.5). Roughly stated, σ is the obstruction to capping " with a boundary and ¿ then classi¯es the di®erent ways of ¯tting a boundary on.