Contents: 0. Review of Surgery Theory. 1. The Low Dimensional Results. 2. Calculation of Normal Maps. 3. Surgery Theory. 4. Computation of Stable Structure Sets. 5. A Construction of Novikov, Cochran & Habegger. 6. Examples. 7. The Topological Case in General. 8. The Smooth Case in Dimension 4. x0. Review of Surgery Theory. Surgery theory is a method for constructing manifolds satisfying a given collection of homotopy conditions. It is usually combined with the s--cobordism theorem which constructs homeomorphisms or diffeomorphisms between two similar looking manifolds. Building on work of Sullivan, Wall applied these two techniques to the problem of computing structure sets. While this is not the only use of surgery theory, it is the aspect on which we will concentrate in this survey. In dimension 4, there are two versions, one in which one builds topological manifolds and homeomorphisms and the second in which one builds smooth manifo

In his 1974 thesis, Martin Scharlemann constructed a fake homotopy equivalence from a closed smooth manifold f : Q # S 3 S 1 #S 2 S 2 , and asked the question whether or not the manifold Q itself is di#eomorphic to S 3 S 1 #S 2 S 2 . Here we answer this question a#rmatively. In [Sc] Scharlemann showed that if # 3 is the Poincare homology 3-sphere, by surgering the 4-manifold # S 1 , along a loop in # 1 # # S 1 normally generating the fundamental group of #, one obtains a closed smooth manifold Q and homotopy equivalence: f : Q -# S 3 S 1 #S 2 S 2 which is not homotopic to a di#eomorphism (actually by taking # to be any Rohlin invariant 1 homology sphere one gets the same result). He then posed the question whether Q is a standard copy of S 3 S 1 #S 2 S 2 (i.e. whether f is a fake self-homotopy equivalence) or Q itself is a fake copy of S 3 S 1 #S 2 S 2 . This question has stimulated much research during the pas...

this paper we make the following two conjectures, relate them to standard conjectures in manifold theory, and thereby prove the following two conjectures for large classes of fundamental groups. Conjecture 0.1 If M and N are closed, orientable, smooth 4-manifolds which are homotopy equivalent and have torsion-free fundamental group, then they are stably di#eomorphic. Conjecture 0.2 If M and N are closed, orientable, topological 4-manifolds which are homotopy equivalent, have torsion-free fundamental group, and have the same KirbySiebenmann invariant, then they are stably homeomorphic. In the simply-connected case the validity of Conjecture 0.1 is well-known by the work of Wall [W1], who showed that homotopy equivalent, smooth, simply-connected 4-manifolds are h-cobordant, and that h-cobordant, smooth, simply-connected manifolds are stably di#eomorphic. Using gauge theory, Donaldson [D] showed that they need not be di#eomorphic. In the simply-connected case, Conjecture 0.2 follows from the work of Freedman [F], with the stronger conclusion that the manifolds are actually homeomorphic. P. Teichner in his thesis [T1] constructed an example of two closed, orientable, # Supported by the Alexander von Humboldt-Stiftung and the National Science Foundation. The author wishes to thank the Johannes Gutenberg-Universitat in Mainz for its hospitality while this work was carried out. 1 homotopy equivalent, smooth 4-manifolds with finite fundamental group which are not stably di#eomorphic. There is a map # 2 : H 2 (#; Z 2 ) # L 4 (Z#), which appears in the surgery classification of high-dimensional manifolds. (Here L = L h , and refers to the Witt group of quadratic forms on free Z#-modules.) As we shall see, this map is conjectured to be injective for all torsion-free groups...

this paper is to extend the above theory to finitely dominated Poincare complexes, that is Poincare complexes in the sense of Wall [18], and to the Witt group L (#) of quadratic structures on f. g. projective Z[#]-modules introduced by Novikov [8], the groups denoted by U (Z[#]) in Ranicki [12]. A normal map (f,b):M X from a compact n-dimensional manifold M to a finitely dominated PoincarecomplexX has a normal bordism invariant, the "projective surgery obstruction" n (# 1 (X)), such that ((f, b) 1:M )=(0,# (f,b)) n+1 (# 1 (X )) = L n+1 (# 1 (X)) n (# 1 (X)). Thus a finitely dominated n-dimensional PoincarecomplexX (n 4) has XS homotopy equivalent to a compact (n + 1)-dimensional CAT manifold if and only if # X admits a CAT reduction for which the corresponding normal map (f,b):M X has projective surgery Partially supported by the Danish Natural Science Research Council. Partially supported by the NSF Grant MCS 79-02017 This paper was published in Topology 19, 239--254 (1980) . n (# 1 (X)). The point we are making here is that the BrowderNovikov transversality construction of normal maps from CAT reductions of # X applies equally well to finitely dominated Poincare complexes X

Abstract. We classify, up to homeomorphism, all closed manifolds having the homotopy type of a connected sum of two copies of real projective n-space. 1. Statement of results Let P n = Pn(R) be real projective n-space. López de Medrano [LdM71] and C.T.C. Wall [Wal68, Wal99] classified, up to PL homeomorphism, all closed PL manifolds homotopy equivalent to P n when n> 4. This was extended to the topological category by Kirby-Siebenmann [KS77, p. 331]. Four-dimensional surgery [FQ90] extends the homeomorphism classification to dimension 4. Cappell [Cap74a, Cap74c, Cap76] discovered that the situation for connected sums is much more complicated. In particular, he showed [Cap74b] that there are closed manifolds homotopy equivalent to P 4k+1 #P 4k+1 which are not nontrivial connected sums. Recent computations of the unitary nilpotent group for the integers by Connolly-Ranicki [CR05], Connolly-Davis [CD04], and Banagl-Ranicki [BR06] show that there are similar examples in dimension 4k (see [JK] for an analysis when k = 1).

Dedicated to Robion Kirby on the occasion of his 60 th birthday In his 1974 thesis, Martin Scharlemann constructed a fake homotopy equivalence from a closed smooth manifold f: Q → S 3 × S 1 #S 2 × S 2, and asked the question whether or not the manifold Q itself is diffeomorphic to S 3 × S 1 #S 2 × S 2. Here we answer this question affirmatively. In [Sc] Scharlemann showed that if Σ 3 is the Poincaré homology 3-sphere, by surgering the 4-manifold Σ × S 1, along a loop in Σ × 1 ⊂ Σ × S 1 normally generating the fundamental group of Σ, one obtains a closed smooth manifold Q and homotopy equivalence: f: Q − → S 3 × S 1 #S 2 × S 2 which is not homotopic to a diffeomorphism (actually by taking Σ to be any Rohlin invariant 1 homology sphere one gets the same result). He then posed the question whether Q is a standard copy of S 3 ×S 1 #S 2 ×S 2 (i.e. whether f is a fake self-homotopy equivalence) or Q itself is a fake copy of S 3 ×S 1 #S 2 ×S 2.

There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly well-developed theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental

Abstract. This is a short survey of some connections between the intersection form and the fundamental group for smooth and topological 4-manifolds. 1.

Abstract. Closed, connected, nonorientable, topological 4-manifolds with infinite cyclic fundamental group are classified. The classification is an extension of results of Freedman and Quinn and of Kreck. The stable classification of such 4-manifolds is also obtained. In principle, it is a consequence of Freedman’s work on topological 4-manifolds that one can classify topological 4-manifolds with “good ” fundamental group. Roughly, a “good ” group is a group for which topological surgery in dimension 4 works ([F2]). For example, by results of Freedman, all groups of polynomial growth are “good”. However, the full classification for closed, orientable 4-manifolds has only been obtained for manifolds with cyclic fundamental groups ([F1][FQ], [HK1][HK2], [FQ][K][SW][Wa]). For closed, nonorientable 4-manifolds the classification has only been given for fundamental group Z2 ([HKT]). In this note, we give the classification for closed, nonorientable 4-manifolds with infinite cyclic fundamental group. Our main result is the following theorem: Theorem 1. (1) Existence: Suppose (H, λ) is a nonsingular ω1-hermitian form on a finitelygenerated free Z[Z]-module, k ∈ Z2, and if λ is even then we assume k =[λ] ∈ L4(Z −). Then there is a closed, connected, nonorientable 4-manifold with π1 = Z, intersection form λ and Kirby-Siebenmann invariant k. (2) Uniqueness: Suppose M and N are closed 4-manifolds with π1 = Z, not orientable but locallyoriented, h: H2(M; Z[Z]) − → H2(N; Z[Z]) is a Z[Z]-isomorphism which preserves intersection forms, and ks(M) = ks(N). Then there is a homeomorphism f: M − → N which induces the identification of fundamental groups, preserves local orientations, and with f ∗ = h.

Abstract. Under certain homological hypotheses on a compact 4-manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. This technical result allows an extension of Cappell’s 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Typically, these homotopy fibers have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be