also appears in this context. Following a suggestion by Chris Woodward, we introduce a “twisted” version $HSI(Y,c)$ of this invariant, associated to a 3-manifold $Y$ endowed with a class $c$ in $H_1(Y,{{\mathbb{Z}}_{2}})$, or equivalently an isomorphism class of $SO(3)$-bundles over $Y$.

Statement of the main results
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Concerning connected sum, we obtain the following Künneth formula:

(Künneth formula for connected sum)\[sommecnx\] Let $Y$ and $Y'$ be two closed oriented 3-manifolds, and $c, c'$ two classes in $H_1(Y;{{\mathbb{Z}}_{2}})$ and $H_1(Y';{{\mathbb{Z}}_{2}})$ respectively. Then, $$\begin{aligned}
HSI(Y \# Y' , c+ c') \simeq & HSI(Y, c) \otimes HSI(Y' , c' )  \\
 & \oplus \mathrm{Tor}(HSI(Y, c) , HSI(Y' , c' ))[-1].\end{aligned}$$

Concerning Dehn surgery, recall first the definition of a surgery triad:

\[def:triad\] A *surgery triad* is a triple of 3-manifolds $Y_\alpha$, $Y_\beta$ and $Y_\gamma$ obtained from a compact oriented 3-manifold $Y$ with genus one boundary by gluing a solid torus along the boundary, identifying the meridian with respectively three simple curves $\alpha$, $\beta$ and $\gamma$, such that $\alpha. \beta = \beta. \gamma =\gamma. \alpha = -1$.

Our exact sequence can then be stated as:

\[suiteintro\]\[trianglechir\] Let $(Y_\alpha , Y_\beta, Y_\gamma)$ be a surgery triad obtained from $Y$ as in the previous definition, $c\in 