Exact triangles for \(SO(3)\) instanton homology of webs (Q2826645)

The present paper is concerned with computational aspects of instanton homology for webs -- an invariant that has been developed by the authors departing from the now classical ideas of instanton Floer homology over manifolds. Here a web is an extention of the notion of a link, obtained by allowing vertices at which three arcs meet along distinct tangent directions. In fact, in order to establish a relationship to the theory of instantons, an alternative definition is useful (see Section 2 of the paper): a web \(K\) is the set of those points in the underlying topological space \(X\) of a three-dimensional bifold (a specific type of orbifold) \(\check{X}\), whose local stabilizer subgroup is non-trivial. The invariant \(J(X,k;\mu)\) the authors are interested in arises as the Floer homology of the (perturbed) Chern-Simons functional on the space of connections (satisfying some local constraint and together with a suitably defined marking \(\mu\) used to cut down the automorphism group) on an \(SO(3)\)-orbibundle over \(\check{X}\). In the case of a web \(K\) in \(\mathbb{R}^{3}\), there is a natural way of how to compactify \(K\) to a web \(K^{\#}\) in (marked) \(S^{3}\) and one sets \(J^{\#}(K)=J(S^{3},K^{\#};\mu)\).NEWLINENEWLINEThe main result of the paper (Theorem 1.1) establishes a long exact sequence that explains the behaviour of the invariant \(J^{\#}(K)\) under a (cyclic) elementary manipulation of the web \(K\) along a pair of adjacent vertices. Both the Theorem and its proof are motivated by analogous results previously obtained by the same authors in the case of \(SU(2)\)-instanton homology for links. In more detail, the maps forming the exacts sequence are obtained as the homomorphisms on Floer homology associated to certain orbifold cobordisms corresponding to the elementary moves that are being considered. In order to prove that the composition of any two consequent maps in the sequence vanishes, it is shown that the cobordism underlying this composition is obtained from a trivial cobordism by a attaching a certain explicitly given piece (see Section 5). The claim then follows using a neck stretching argument by studying the holonomy action on connections over this additional piece (Section 4). Exactness of the sequence is proved using another neck stretching argument.NEWLINENEWLINEThe last section of the paper is concerned with applications. The authors had previously conjectured that in the case when the web \(K\) is planar, the total dimension of \(J^{\#}(K)\) is equal to the number of Tait colorings of the abstract graph underlying \(K\) and showed that this conjecture would imply the famous four-color theorem. In turn, it is proved here that this previous conjecture of the authors is equivalent to a certain formula for the behaviour of the total dimension of \(J^{\#}(K)\) under elementary manipulations of the web \(K\). Some partial headway into verifying this latter formula is made.