# An instanton-invariant for 3-manifolds

@article{Floer1988AnIF, title={An instanton-invariant for 3-manifolds}, author={Andreas Floer}, journal={Communications in Mathematical Physics}, year={1988}, volume={118}, pages={215-240} }

To an oriented closed 3-dimensional manifoldM withH1(M, ℤ)=0, we assign a ℤ8-graded homology groupI*(M) whose Euler characteristic is twice Casson's invariant. The definition uses a construction on the space of instantons onM×ℝ.

#### 439 Citations

Instantons for 4-manifolds with periodic ends and an obstruction to embeddings of 3-manifolds

- Mathematics
- Topology and its Applications
- 2018

Abstract We construct an obstruction to the existence of embeddings of a homology 3-sphere into a homology S 3 × S 1 under some cohomological condition. The obstruction is defined as an element in… Expand

Four-manifold invariants from higher-rank bundles

- Mathematics
- 2004

Generalized Donaldson invariants of 4-manifolds are defined, using moduli spaces of anti-self-dual connections with structure group SU(N) or PSU(N). Some values of the invariants are calculated for… Expand

An inequality for the h -invariant in instanton Floer theory

- Mathematics
- 2001

Abstract Given a smooth, compact, oriented 4-manifold X with a homology sphere Y as boundary and b 2 + ( X )=1, and given an embedded surface Σ ⊂ X of self-intersection 1, we prove an inequality… Expand

Non-trivial involutions on Floer Homology

- Mathematics
- 2003

Abstract. We construct the first examples of irreducible 3-manifolds with the homology of S1×S2 admitting an involution acting non-trivially on their Floer Homology. The examples are obtained by… Expand

Lescop's Invariant and Gauge Theory

- Mathematics
- 2013

Taubes proved that the Casson invariant of an integral homology 3-sphere equals half the Euler characteristic of its instanton Floer homology. We extend this result to all closed oriented 3-manifolds… Expand

Knot homology groups from instantons

- Mathematics
- 2011

For each partial flag manifold of SU(N), we define a Floer homology theory for knots in 3-manifolds, using instantons with codimension-2 singularities.

Chern-Simons invariants, SO(3) instantons, and Z/2-homology cobordism

- Mathematics
- 2010

We review the SO(3) instanton gauge theory of Fintushel and Stern and recast it in the context of 4-manifolds with cylindrical ends. Appli- cations to the Z/2-homology cobordism group of Z/2-homology… Expand

Mutation and gauge theory I: Yang-Mills invariants

- Mathematics
- 1997

Abstract. Mutation of 3-manifolds (cutting and regluing along a genus 2 surface using a central involution) is shown to preserve the instanton Floer homology of homology 3-spheres. A related… Expand

Toroidal homology spheres and SU(2)-representations

- Mathematics
- 2021

We prove that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)representations. Our methods use instanton Floer… Expand

A PRODUCT FORMULA OF SEIBERG-WITTEN INVARIANTS

- Mathematics
- 2003

Let X be a 4-manifold obtained by gluing two symplectic 4-manifolds Xi, i = 1, 2, along embedded surfaces. Using the gradient flow of a functional on 3-dimensional Seiberg-Witten theory along the… Expand

#### References

SHOWING 1-10 OF 28 REFERENCES

Instantons and geometric invariant theory

- Mathematics
- 1984

We show that the Yang-Mills instantons can be described in terms of certain holomorphic bundles on the projective plane. The proof uses explicit matrix descriptions arising from monads and an… Expand

Instantons and Four-Manifolds

- Mathematics
- 1984

This volume has been designed to explore the confluence of techniques and ideas from mathematical physics and the topological study of the differentiable structure of compact four-dimensional… Expand

Gauge theory on asymptotically periodic {4}-manifolds

- Mathematics
- 1987

Il existe une famille non denombrable de classes de diffeomorphismes des 4-varietes orientees homeomorphes a R 4

Connections withLP bounds on curvature

- Mathematics
- 1982

We show by means of the implicit function theorem that Coulomb gauges exist for fields over a ball inRn when the integralLn/2 field norm is sufficiently small. We then are able to prove a weak… Expand

The unregularized gradient flow of the symplectic action

- Mathematics
- 1988

The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of this… Expand

Removable singularities in Yang-Mills fields

- Mathematics
- 1982

We show that a field satisfying the Yang-Mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite. We obtain the result that every… Expand

Symplectic fixed points and holomorphic spheres

- Mathematics
- 1989

LetP be a symplectic manifold whose symplectic form, integrated over the spheres inP, is proportional to its first Chern class. On the loop space ofP, we consider the variational theory of the… Expand

Construction of Instantons

- Physics
- 1978

A complete construction, involving only linear algebra, is given for all self-dual euclidean Yang-Mills fields.

Topological quantum field theory

- Physics
- 1988

A twisted version of four dimensional supersymmetric gauge theory is formulated. The model, which refines a nonrelativistic treatment by Atiyah, appears to underlie many recent developments in… Expand

Morse theory for Lagrangian intersections

- Mathematics
- 1988

Soit P une variete symplectique compacte et soit L⊂P une sous-variete lagrangienne avec π 2 (P,L)=0. Pour un diffeomorphisme exact φ de P avec la propriete que φ(L) coupe L transversalement, on… Expand