Fukumoto–Furuta invariants of plumbed homology 3-spheres

Saveliev, Nikolai

doi:10.2140/pjm.2002.205.465

Cited by 40 publications

(89 citation statements)

References 16 publications

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“…On the other hand, Fukumoto and Furuta (see [18][19]) defined certain invariants for plumbed homology 3-spheres using gauge theory. According to a recent result of Saveliev [100], these invariants fit together to give a candidate of another homomorphism on Θ …”

Section: Group Of Homology Cobordism Classes Of Homology Cylindersmentioning

confidence: 93%

Cohomological structure of the mapping class group and beyond

Morita¹

2006

Proceedings of Symposia in Pure Mathematics

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Abstract. In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of free groups, the symplectomorphism group of surfaces.Based on these, we present several conjectures and problems concerning the cohomology of these groups. We are particularly interested in the possible interplays between these cohomology groups rather than merely the structures of individual groups. It turns out that, we have to include, in our considerations, two other groups which contain the mapping class group as their core subgroups and whose structures seem to be deeply related to that of the mapping class group. They are the arithmetic mapping class group and the group of homology cobordism classes of homology cylinders.

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Section: Group Of Homology Cobordism Classes Of Homology Cylindersmentioning

confidence: 93%

Cohomological structure of the mapping class group and beyond

Morita¹

2006

Proceedings of Symposia in Pure Mathematics

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“…In Figure 3 the Euler numbers and form a straight line or chain structure. We associate to each node a weight equal to zero that is connected with using of the unnormalized Seifert invariants for Bh-spheres, which are the block elements for the construction of graph manifolds [6]. Now let's define a Laplacian matrix for the plumbing graph p Γ as follows:…”

Section: Block Matrix Representation For a Graph P γ Of Tree Typementioning

confidence: 99%

Block Matrix Representation of a Graph Manifold Linking Matrix Using Continued Fractions

Lopez¹,

Ефремов²,

Magdaleno³

2014

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We consider the block matrices and 3-dimensional graph manifolds associated with a special type of tree graphs. We demonstrate that the linking matrices of these graph manifolds coincide with the reduced matrices obtained from the Laplacian block matrices by means of Gauss partial diagonalization procedure described explicitly by W. Neumann. The linking matrix is an important topological invariant of a graph manifold which is possible to interpret as a matrix of coupling constants of gauge interaction in Kaluza-Klein approach, where 3-dimensional graph manifold plays the role of internal space in topological 7-dimensional BF theory. The Gauss-Neumann method gives us a simple algorithm to calculate the linking matrices of graph manifolds and thus the coupling constants matrices.

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“…As the referee pointed out, one can also use the orbifold version of Furuta's Theorem [11] and arguments similar to those in [22] to prove Corollary 2 as well as the claims made in the last column of Table 1.…”

Section: Remarkmentioning

confidence: 99%

“…It is for instance a consequence of Donaldson's famous theorem about the intersection forms of smooth 4-manifolds that the Poincaré homology sphere Σ(2, 3, 5) has infinite order in this group, and M. Furuta found a family of Brieskorn spheres which generates a subgroup of infinite rank [12]. Recently N. Saveliev [22] showed, using the w-invariant introduced in [11], that a Brieskorn sphere with non-trivial Rokhlin invariant has infinite order in the integral homology cobordism group. However, many 3-manifolds arising naturally in knot theory, for instance double coverings of the 3-sphere branched along knots, are not integral homology spheres, but still Z 2 -homology spheres.…”

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confidence: 99%

Homology cobordism and classical knot invariants

Bohr¹,

Lee²

2002

Commentarii Mathematici Helvetici

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Abstract. In this paper, we define and investigate Z 2 -homology cobordism invariants of Z 2 -homology 3-spheres which turn out to be related to classical invariants of knots. As an application, we show that many lens spaces have infinite order in the Z 2 -homology cobordism group and we prove a lower bound for the slice genus of a knot on which integral surgery yields a given Z 2 -homology sphere. We also give some new examples of 3-manifolds which cannot be obtained by integral surgery on a knot. Mathematics Subject Classification (2000). 57M27, 57M25.Keywords. Homology 3-spheres, homology cobordism, slice genus.In recent years, gauge theoretical tools and new results in 4-dimensional topology have successfully been used to study the structure of the integral homology cobordism group. It is for instance a consequence of Donaldson's famous theorem about the intersection forms of smooth 4-manifolds that the Poincaré homology sphere Σ(2, 3, 5) has infinite order in this group, and M. Furuta found a family of Brieskorn spheres which generates a subgroup of infinite rank [12]. Recently N. Saveliev [22] showed, using the w-invariant introduced in [11], that a Brieskorn sphere with non-trivial Rokhlin invariant has infinite order in the integral homology cobordism group. However, many 3-manifolds arising naturally in knot theory, for instance double coverings of the 3-sphere branched along knots, are not integral homology spheres, but still Z 2 -homology spheres. As in the case of integral homology spheres, the set of Z 2 -homology spheres modulo the Z 2 -homology cobordism relation forms a group, the so called Z 2 -homology cobordism group Θ 3 Z 2 . To study this group, we introduce, based on Furuta's result on the intersection forms of smooth 4-dimensional spin manifolds [13], two invariants of Z 2 -homology spheres which turn out to be in fact invariants of the cobordism class (see Theorem 1). Exploiting that these invariants are closely related to classical knot invariants like signature and slice genus, we prove estimates for them in the case of lens spaces, which enables us to exhibit many examples of lens spaces which have infinite order inThe first author gratefully acknowledges support from the Deutsche Forschungsgemeinschaft.

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Fukumoto–Furuta invariants of plumbed homology 3-spheres

Cited by 40 publications

References 16 publications

Cohomological structure of the mapping class group and beyond

Cohomological structure of the mapping class group and beyond

Block Matrix Representation of a Graph Manifold Linking Matrix Using Continued Fractions

Homology cobordism and classical knot invariants

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