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A Survey of 4Manifolds Through the Eyes of Surgery.
"... 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 Dimensio ..."
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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 scobordism 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
Scharlemann’s manifold is standard
"... 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 ..."
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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 Poincare homology 3sphere, by surgering the 4manifold 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. This question has stimulated much research during the past twenty years resulting in some partial answers. For example, in [FP] it was shown that Q is stably standard, in [Sa] it was proven that it is obtained by surgering a knotted S 2 ⊂ S 2 ×S 2, and in [A3] it was shown that it is obtained by “Gluck twisting ” S 3 × S 1 #S 2 × S 2 along an imbedded 2sphere. Also by a similar construction one can obtain a fake homotopy equivalence: g: P − → S 3 ˜×S 1 #S 2 × S 2 where S 3 ˜×S 1 is the twisted S 3 bundle over S 1. But in this case it turns out
The Borel/Novikov conjectures and stable diffeomorphisms of 4manifolds
"... 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 4manifolds which are homotopy equivalent and ha ..."
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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 4manifolds which are homotopy equivalent and have torsionfree fundamental group, then they are stably di#eomorphic. Conjecture 0.2 If M and N are closed, orientable, topological 4manifolds which are homotopy equivalent, have torsionfree fundamental group, and have the same KirbySiebenmann invariant, then they are stably homeomorphic. In the simplyconnected case the validity of Conjecture 0.1 is wellknown by the work of Wall [W1], who showed that homotopy equivalent, smooth, simplyconnected 4manifolds are hcobordant, and that hcobordant, smooth, simplyconnected manifolds are stably di#eomorphic. Using gauge theory, Donaldson [D] showed that they need not be di#eomorphic. In the simplyconnected 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 HumboldtStiftung and the National Science Foundation. The author wishes to thank the Johannes GutenbergUniversitat in Mainz for its hospitality while this work was carried out. 1 homotopy equivalent, smooth 4manifolds 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 highdimensional 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 torsionfree groups...
MANIFOLDS HOMOTOPY EQUIVALENT TO P n #P n
, 2006
"... Abstract. We classify, up to homeomorphism, all closed manifolds having the homotopy type of a connected sum of two copies of real projective nspace. 1. Statement of results Let P n = Pn(R) be real projective nspace. López de Medrano [LdM71] and C.T.C. Wall [Wal68, Wal99] classified, up to PL home ..."
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Abstract. We classify, up to homeomorphism, all closed manifolds having the homotopy type of a connected sum of two copies of real projective nspace. 1. Statement of results Let P n = Pn(R) be real projective nspace. 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 KirbySiebenmann [KS77, p. 331]. Fourdimensional 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 ConnollyRanicki [CR05], ConnollyDavis [CD04], and BanaglRanicki [BR06] show that there are similar examples in dimension 4k (see [JK] for an analysis when k = 1).
PROJECTIVE SURGERY THEORY
"... 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] ..."
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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 ndimensional 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 ndimensional 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 7902017 This paper was published in Topology 19, 239254 (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
INTERSECTION FORMS, FUNDAMENTAL GROUPS AND 4MANIFOLDS
"... Abstract. This is a short survey of some connections between the intersection form and the fundamental group for smooth and topological 4manifolds. 1. ..."
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Abstract. This is a short survey of some connections between the intersection form and the fundamental group for smooth and topological 4manifolds. 1.
Scharlemann’s manifold is standard
, 1999
"... 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 ..."
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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 3sphere, by surgering the 4manifold 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 selfhomotopy equivalence) or Q itself is a fake copy of S 3 ×S 1 #S 2 ×S 2.
BETWEEN LOWER AND HIGHER DIMENSIONS (in the work of Terry Lawson)
"... 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 ..."
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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 welldeveloped 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
CLASSIFICATION OF CLOSED NONORIENTABLE 4MANIFOLDS WITH INFINITE CYCLIC FUNDAMENTAL GROUP
, 1995
"... Abstract. Closed, connected, nonorientable, topological 4manifolds 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 4manifolds is also obtained. In principle, it is a consequen ..."
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Abstract. Closed, connected, nonorientable, topological 4manifolds 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 4manifolds is also obtained. In principle, it is a consequence of Freedman’s work on topological 4manifolds that one can classify topological 4manifolds 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 4manifolds has only been obtained for manifolds with cyclic fundamental groups ([F1][FQ], [HK1][HK2], [FQ][K][SW][Wa]). For closed, nonorientable 4manifolds the classification has only been given for fundamental group Z2 ([HKT]). In this note, we give the classification for closed, nonorientable 4manifolds with infinite cyclic fundamental group. Our main result is the following theorem: Theorem 1. (1) Existence: Suppose (H, λ) is a nonsingular ω1hermitian 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 4manifold with π1 = Z, intersection form λ and KirbySiebenmann invariant k. (2) Uniqueness: Suppose M and N are closed 4manifolds 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.
ON SURGERY FOR 4MANIFOLDS AND SPLITTING OF 5MANIFOLDS
, 2007
"... Abstract. Under certain homological hypotheses on a compact 4manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. This technical result allows an extension of Cappell’s 5dimensional splitting theorem. As an application, we analyze, up to inte ..."
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Abstract. Under certain homological hypotheses on a compact 4manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. This technical result allows an extension of Cappell’s 5dimensional splitting theorem. As an application, we analyze, up to internal scobordism, the splitting and fibering problems for certain 5manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4manifolds. Typically, these homotopy fibers have nonvanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be