We compute the involutive Heegaard Floer homology of the family of threemanifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involutive correction terms in the case that all of the summands have the same orientation. Using these calculations, we give a new proof of the existence of an infinite-rank subgroup in the three-dimensional homology cobordism group.Theorem 1.2. Let Y and s be as in Theorem 1.1. Then, the involutive Heegaard Floer correction terms are given by dpY, sq "´2μpY, sq,dpY, sq " dpY, sq, where dpY, sq is the Ozsváth-Szabó d-invariant andμpY, sq is the Neumann-Siebenmann invariant from [17], [23]. Theorems 1.1 and 1.2 should be compared with the corresponding results for Pinp2q-monopole Floer homology, obtained by the first author in [5]. For the smaller class of rational homology spheres Seifert fibered over a base orbifold with underlying space S 2 , the Pinp2q-equivariant Seiberg-Witten Floer homology was computed by Stoffregen in [25].The proof of Theorem 1.1 is in two steps. First, we identify the action of J 0 on H´pR k q (or, more precisely, H´pR k qrσ`2sq with the conjugation involution on HF´pY, sq, using an argument from [5]. Second, we prove that, in the case at hand, the action of J 0 on H´pR k q 1 In this paper, by ras we will denote a grading shift by a, so that an element in degree 0 in a module M becomes an element in degree´a in the shifted module M ras. This is the standard convention, but opposite to the one in [14].

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We establish a structural understanding of the involutive Heegaard Floer homology for all linear combinations of almost-rational (AR) plumbed three-manifolds. We use this to show that the Neumann-Siebenmann invariant is a homology cobordism invariant for all linear combinations of AR plumbed homology spheres. As a corollary, we prove that if Y is a linear combination of AR plumbed homology spheres with µpY q " 1, then Y is not torsion in the homology cobordism group. A general computation of the involutive Heegaard Floer correction terms for these spaces is also included.2010 Mathematics Subject Classification. 57R58, 57M27.

Building on the algebraic framework developed by Hendricks, Manolescu, and Zemke, we introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution over any homology ball, rather than a particular contractible manifold. Unlike previous approaches, we do not utilize any closed 4-manifold topology or contact topology. Instead, we adapt the formalism of local equivalence coming from involutive Heegaard Floer homology. As an application, we define a modification Θ τ Z of the homology cobordism group which takes into account an involution on each homology sphere, and prove that this admits a Z 8 -subgroup of strongly non-extendable corks. The group Θ τ Z can also be viewed as a refinement of the bordism group of diffeomorphisms. Using our invariants, we furthermore establish several new families of corks and prove that various known examples are strongly non-extendable. Our main computational tool is a monotonicity theorem which constrains the behavior of our invariants under equivariant negative-definite cobordisms, and an explicit method of constructing such cobordisms via equivariant surgery.

We compute the Pin(2)-equivariant monopole Floer homology for the class of plumbed 3-manifolds considered by Ozsváth and Szabó in [18]. We show that for these manifolds, the Pin(2)-equivariant monopole Floer homology can be calculated in terms of the Heegaard Floer/monopole Floer lattice complex defined by Némethi [15]. Moreover, we prove that in such cases the ranks of the usual monopole Floer homology groups suffice to determine both the Manolescu correction terms and the Pin(2)-homology as an abelian group. As an application of this, we show that βp´Y, sq "μpY, sq for all plumbed 3-manifolds with at most one "bad" vertex, proving (an analogue of) a conjecture posed by Manolescu in [12]. Our proof also generalizes results by Stipsicz [21] and Ue [26] relatingμ with the Ozsváth-Szabó d-invariant. Some observations aimed at extending our computations to manifolds with more than one bad vertex are included at the end of the paper.