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Uniqueness of Connected Sum Decompositions in Dimension 4

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Uniqueness of Connected Sum Decompositions in Dimension 4 View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector TOPOLOGY AND ITS APPLICATIONS EISEVIER Topology and its Applications 56 (1994) 277-291 Uniqueness of connected sum decompositions in dimension 4 Richard Stong Department of Mathematics, University of California, Los Angeles, CA 90024-1555, USA (Received 9 November 1992; revised 16 March 1993) Abstract Suppose W is a 4-manifold with good fundamental group and M is a closed simply-con- nected 4-manifold. Suppose we are given two decompositions h, : WY M#W, and h,: W = M#W, inducing the same decomposition of n-*W. In this paper we study when we can conclude that WI and W, are homeomorphic. As a consequence we conclude that the * operation for changing the Kirby-Siebenmann invariant of a 4-manifold is well defined. We will also use this discussion to relate the ambient approach to classification to the surgery approach. Key words: 4-manifolds AMS CMOS) Subj. Class.: 57N13 1. Introduction It is well known that topological 4-manifolds do not have the same sort of uniqueness of connected sum decompositions that 3-manifolds do. For example, one has the two decompositions CP*#(@P*# - @P2) E CP2#(S2 X S*>. This example is constructed by realizing two different decompositions of r2 as con- nected sum decompositions. Therefore to obtain any sort of uniqueness result we must look at only connected sum decompositions inducing the same decomposition of r2. This alone is not sufficient since we still have the example CP*#CP* = * CP2# *@P, where *CP2 is the Chern manifold, the homotopy @P2 with nonzero Kirby-Siebenmann invariant. This example is built by splitting the Kirby-Siebenmann invariant differently between the factors. One way to remove this example is to further fix one of the factors. Also as a technical convenience we will insist that this factor be simply connected (this restriction is easy to remove as is discussed below). 0166~8641/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved &SD1 0166-8641(93)E0067-X 278 R. Stong / Topology and its Applications 56 (1994) 277-291 Accordingly throughout this paper let W be a 4-manifold with good fundamen- tal group r = r,W and M a simply-connected 4-manifold. For any 4-manifold X let X0 denote X with an open ball removed. Suppose that we have a connected sum decomposition h, : W = M#W,, then we get a decomposition r,W = (r,M &I Zr) f3 rzWl. We will say M is s-characteristic in W (s standing for spherically) if the second Stiefel-Whitney class o2 viewed as a map o2 : r,W + 2/2 does not vanish but w z : rTT2Wl--f 2/2 does vanish. Suppose we have another connected sum decomposition h, : W = M#W, inducing the same decomposition of r,W. We wish to understand whether WI and W, are homeomorphic. For example, part of the uniqueness portion of Theorem 10.3 of Freedman and Quinn [41 is the fact that if M is even, then the complements W, and W, are homeomorphic (and the homeomorphism is homotopic to the canonical homotopy equivalence). The main goal of this paper is to prove a partial extension of this result to the case where A4 is odd. The obstructions to the existence of a pseudoisotopy between these decomposi- tions (that is, the existence of a homeomorphism W X I --) W X I which is the identity on W x (0) and on W X 11) it is a homeomorphism g : W + W with h, 0 g o(h,jpl the identity on M,) are given in [4, ch. 101 and [7]. There is unfortunately a slight subltety in defining these invariants. Fix a homotopy H from h,toh,,whichwewillviewasahomotopyH:M,XZ-,WXZbetween(h,)-‘l~u, and (h,)-‘IM,,. The obstructions are most easily defined as obstructions to finding a pseudoisotopy whose restriction to (h,)-‘CM, x Z) agrees with H up to homol- ogy with Z[ r]-coefficients. Therefore the obstructions depend on the homotopy H. Since most of our conclusions depend only on h, and h, it is possible to write down more complicated statements without this dependence. The first obstruction is the selfintersection obstruction fq(H) defined in [4, p. 1841 (with minor corrections in [7]). Let T, be the elements of order two in r,W on which the first Stiefel-Whitney class w1 vanishes (and T_ those with w, nonzero). The selfintersection obstruction is essentially the selfintersection form (of the 6-manifold W X Z X R> applied to H. It takes values in hom,(H,(M; Zr), Z/2[T+]) = H2(M; 2/2[T+I). If M is not s-characteristic, then all these values will be attained for some choice of h, and H. If A4 is s-characteris- tic, then there is a subgroup G c H2(M; 2/2[T+I) of index at most two (defined below) such that fq(H) takes on exactly the values in G. If M is not s-characteris- tic, then fq(H) is the only obstruction. If A4 is s-characteristic, then there is a secondary obstruction km(H) E H,(W; 2/2)/r, which is a 5-dimensional analog of the Kervaire-Milnor obstruction and Z is a subgroup of H,(W; Z/2) defined below. The two subgroups G and Z mentioned above are defined as follows. Assume M is s-characteristic. Let CYE H,(M; Z) be characteristic and let p be the image of (Y in H,(W, z/2). Let g E T+U T_. Let f : RP2 --) W be a map representing g, i.e., with f*r,RP2 = (1, g}. Define a map $J : T+U T_+ 2/2 by #4h) = m2(f *TW) + 0 .f*[RP2] E /21/2. This is clearly independent of the choice of (Y R. Stong/ Topolsgy and its App&ations 56 fI994127?-291 279 since only mod 2 intersection numbers with p enter into the formula. Also note that it is independent of f since any other map of lRP* representing g differs from f by an element of a,@‘. The subgroup f is the subgroup of HJW, Z/2) generated by the elements of T_n (6-‘(l). Extend 4 by Iinearity to a map 4 : 2/2[T+] -+ Z/2. Then G is the subgroup of H’(M; @/2)[T+]) consisting of elements LJwith q(a) E ker(#), where (Yis a characteristic class for M as above. In particular if M is characteristic in the usual sense, then 4 = 0 and hence r is trivial and G is a11of H”(M; (Z/2XT+]). A few more facts about 4 can be found in [7]. One central property of these obstructions is their additivity. Specifically sup- pose H is a homotopy between h, and h, and H’ is a homotopy between h, and h,. (Viewed again as homotopies between (hi)-” : MO + W.) Then the union HUH’ can be viewed as a homotopy Ma X I---) WX I between (hl)-l(MO and (!~,>-~l~~ by making H the restriction to MO X [0,1/2] + WX [0,1/2] and H’ the restriction to M, X [l/2,1] + W x [l/2,1]. Since fq is defined by adding up contri- butions from each selfintersection, we clearly have fq(H u H’) = fq(H) + fq(H’). Similarly if fq(H) = fq(H’) = 0, then km(H U H’) = km(H) + km(H’). Using the machinery described above we will prove the following result. Theorem 1.1. S~p~se M is a closed l-connected 4-man~old and W is a 4-manifold with good fundamental group. Suppose h, : W = M# W, and h, : W = M# W, are two decompositions inducing the same decomposition of rr,W and H a homotopy between them. Let a E H,(M; Z) be characteristic. (a) If fq(H#cr) = 0, then there is a homeomo~h~m W, + W, which is homo- topic io an element in the canonical family of homotopy equivalences W, -+ W, (defined below 1. (b) If A4 is not s-characteristic or fq(H)(a) E L/2[T+n ker 41, then there is a homeomorph~m W, + W, which is homotopic to the cunonical family of homotopy equivalences over the 2-skeleton of W,. If M is not simply connected, then much of this theorem still holds. There are two technical complications. First even if a,W, and -rr,M are good, it will not generally be true that .xlW is good. Second the invariants fq and km are not defined in 141or [7] for A4 not l-connected. For the selfintersection invariant this is only a minor complication. Fix a concordance M X I -+ W x I and extend it to a map M X I + W X I X 52. Then intersection with coefficients in Z[lr,W,] are de- fined and hence so is a selfintersection obstruction in H~(~; Z[T+]) where T+ is the orientable elements in rrlW, of order two. The Kervaire-Milnor obstruction km is defined in [71 combinatorially but without assuming M is r,-negligible. It should be possible to extend it to general M. At any rate since km does not enter into the hypotheses above this is at most a pedagogical problem. Another minor complication is that no machinery exists to produce such a decomposition. Assume that W and W’ are s-cobordant 4-manifolds (say with an s-cobordism VI and M’ is a closed 4-manifold.
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