H_1(Y;{{\mathbb{Z}}_{2}})$, and for $\delta \in \lbrace \alpha,\beta,\gamma\rbrace$, $c_\delta \in H_1(Y_\delta;{{\mathbb{Z}}_{2}})$ the class induced from $c$ by the inclusions. Let also $k_\alpha \in H_1(Y_\alpha;{{\mathbb{Z}}_{2}})$ be the class corresponding to the core of the solid torus. Then, there exists a long exact sequence: $$\cdots\rightarrow HSI(Y_\alpha ,c_\alpha+ k_\alpha) \rightarrow HSI(Y_\beta,c_\beta) \rightarrow HSI(Y_\gamma,c_\gamma)\rightarrow \cdots .$$

As in Heegaard Floer theory, one can define maps associated to four-dimensional cobordisms, and show that two of the three mophisms appearing in this sequence are associated to the canonical cobordisms, with appropriate cohomology classes. This has been done (in French) in the author’s PhD [@these Chapter 6], and will be the object of a forthcoming paper [@cobordisms], which should also address the naturality issue.

Applications
------------

The exact sequence of Theorem \[suiteintro\], together with the knowledge of the Euler characteristic of symplectic instanton homology and an observation from Ozsv[á]{}th and Szab[ó]{}, allow one to compute the HSI groups of several manifolds. We will present some of them in section \[sec:applic\], including plumbings, surgeries on knots, and branched double covers.

Organisation of the paper
-------------------------

In section \[sec:fft\], we review Wehrheim and Woodward’s Floer Field theory and adapt it to the framework of HSI. In section \[sec:hsi\] we will 