Instanton Floer homology for connected sums of Poincaré spheres

Li, Weiping

doi:10.1007/s00209-005-0802-0

Math. Z.

2005

DOI: 10.1007/s00209-005-0802-0

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Instanton Floer homology for connected sums of Poincaré spheres

Weiping Li

Abstract: In this paper, we apply the Mayer-Vietoris principle to compute the integer graded Floer homology of the connected sum of a Poincaré homology sphere with itself. We then apply the Fintushel-Stern spectral sequence to deduce its regular Floer homology and observe the presence of 3-torsion in the Floer groups.

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2019

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Seifert hypersurfaces of 2-knots and Chern-Simons functional

Takeuchi¹

2019

Preprint

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We introduce a real-valued functional csK on the SU (2)-representation space of the knot group for any oriented 2-knot K. In order to define the functionals, we use a generalization for homology S 1 × S 3 's of the Chern-Simons functionals introduced in [36]. We calculate our functionals for ribbon 2-knots and the twisted spun 2-knots of torus knots, Montesinos knots and 2-bridge knots. We show several properties of the images of the functionals including a connected sum formula and relationship to the Chern-Simons functionals of Seifert hypersurfaces of K. As a corollary, we show that every oriented 2-knot having an oriented homology 3-sphere of a certain class as its Seifert hypersurface admits an SU (2)-irreducible representation of a knot group. Moreover, we also relate the existence of embeddings from a homology 3-sphere into a negative definite 4-manifold to SU (2)-representations of their fundamental groups. For example, we prove that every closed definite 4-manifold containing Σ(2, 3, 5, 7) as a submanifold has an uncountable family of SU (2)-representations of its fundamental group. This implies that every 2-knot having Σ(2, 3, 5, 7) as a Seifert hypersurface has an uncountable family of SU (2)-representations of its knot group. The proofs of these results use several techniques from instanton Floer theory.

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Seifert hypersurfaces of 2-knots and Chern-Simons functional

Takeuchi¹

2019

Preprint

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Instanton Floer homology for connected sums of Poincaré spheres

Cited by 1 publication

References 13 publications

Seifert hypersurfaces of 2-knots and Chern-Simons functional

Seifert hypersurfaces of 2-knots and Chern-Simons functional

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