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In \cite{MR1957829}, Ozsv\'ath and Szab\'o use Heegaard Floer homology todefine numerical invariants $d_{1/2}$ and $d_{-1/2}$ for 3-manifolds $Y$ with$H_{1}(Y;\mathbb{Z})\cong \mathbb{Z}$. We define involutive Heegaard Floertheoretic versions of these invariants analogous to the involutive $d$invariants $\bar{d}$ and $\underline{d}$ defined for rational homology spheresby Hendricks and Manolescu in \cite{MR3649355} . We prove their invarianceunder spin integer homology cobordism and use them to establish spin fillingconstraints and $0$-surgery obstructions analogous to results by Ozsv\'ath andSzab\'o for their Heegaard Floer counterparts $d_{1/2}$ and $d_{-1/2}$. We thenapply calculation techniques of Dai and Manolescu developed in \cite{MR4021102}and Rustamov in \cite{Rustamov} to compute the involutive Heegaard Floerhomology of some negative semi-definite plumbed 3-manifolds with $b_{1} =1$. Bycombining these calculations with the $0$-surgery obstructions, we are able toproduce an infinite family of small Seifert fibered spaces with weight 1fundamental group and first homology $\mathbb{Z}$ which cannot be obtained by$0$-surgery on a knot in $S^3$, extending a result of Hedden, Kim, Mark, andPark in \cite{MR4029676}.

On the involutive Heegaard Floer homology of negative semi-definite plumbed 3-manifolds with $b_{1}=1$