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Higher order intersection numbers of 2spheres in 4manifolds
 ALGEBRAIC & GEOMETRIC TOPOLOGY
, 2000
"... This is the beginning of an obstruction theory for deciding whether a map f: S2 → X is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres in a topological 4manifold X. The first obstruction is Wall’s well known selfintersection nu ..."
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Cited by 16 (9 self)
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This is the beginning of an obstruction theory for deciding whether a map f: S2 → X is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres in a topological 4manifold X. The first obstruction is Wall’s well known selfintersection number µ(f) which tells the whole story in higher dimensions. Our second order obstruction τ(f) is defined if µ(f) vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of π1X modulo S3symmetry (rather then just one copy modulo S2symmetry). It generalizes to the nonsimply connected setting the KervaireMilnor invariant defined in [2] and [12] which corresponds to the Arfinvariant of knots in 3space. We also give necessary and sufficient conditions for homotoping three maps f1, f2, f3: S2 → X to a position in which they have disjoint images. The obstruction λ(f1, f2, f3) generalizes Wall’s intersection number λ(f1, f2) which answers the same question for two spheres but is not sufficient (in dimension 4) for three spheres. In the same way as intersection numbers correspond to linking numbers in 3space, our new invariant corresponds to the Milnor invariant µ(1, 2, 3), generalizing the Matsumoto triple to the non simplyconnected setting. Finally, we explain some simple algebraic properties of these new cubic forms on π2(X) in Theorem 3. These are straightforward generalizations of the properties of quadratic forms as defined by Wall [14, §5]. A particularly attractive formula is λ(f, f, f) = ∑ τ(f) σ σ∈S3 which generalizes the well known fact that Wall’s invariants satisfy λ(f, f) = µ(f) + µ(f) = ∑ µ(f) σ for an immersion f with trivial normal bundle. σ∈S2 1.
STABILISATION, BORDISM AND EMBEDDED SPHERES
, 2000
"... Abstract. It is one of the most interesting facts in 4–dimensional topology that even in simply–connected 4–manifolds, not every homology class of degree 2 can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that many of the obstructions against constructing such a sph ..."
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Cited by 1 (1 self)
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Abstract. It is one of the most interesting facts in 4–dimensional topology that even in simply–connected 4–manifolds, not every homology class of degree 2 can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that many of the obstructions against constructing such a sphere vanish if one modifies the ambient 4–manifold by adding copies of products of spheres, a process which is usually called stabilisation. In this paper, we extend this result to non–simply connected 4–manifolds and show how it is related to the Spin c –bordism groups of Eilenberg–McLane spaces.
STABILISATION, BORDISM AND EMBEDDED SPHERES IN 4–MANIFOLDS
, 2001
"... Abstract. It is one of the most important facts in 4–dimensional topology that there are 4–manifolds in which not every spherical homology class of degree 2 can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that in the simply connected case, many of the obstructions ..."
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Cited by 1 (0 self)
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Abstract. It is one of the most important facts in 4–dimensional topology that there are 4–manifolds in which not every spherical homology class of degree 2 can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that in the simply connected case, many of the obstructions to constructing such a sphere vanish if one modifies the ambient 4–manifold by adding products of 2–spheres, a process which is usually called stabilisation. In this paper, we extend this result to non–simply connected 4–manifolds and show how it is related to the Spin c – bordism groups of Eilenberg–McLane spaces.
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
Christian Bohr
, 2002
"... Abstract It is one of the most important facts in 4–dimensional topology that not every spherical homology class of a 4–manifold can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that in the simply connected case, many of the obstructions to constructing such a spher ..."
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Abstract It is one of the most important facts in 4–dimensional topology that not every spherical homology class of a 4–manifold can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that in the simply connected case, many of the obstructions to constructing such a sphere vanish if one modifies the ambient 4–manifold by adding products of 2–spheres, a process which is usually called stabilisation. In this paper, we extend this result to non–simply connected 4–manifolds and show how it is related to the Spin c –bordism groups of Eilenberg–MacLane spaces. AMS Classification 57M99; 55N22