On the involutive Heegaard Floer homology of negative semi-definite plumbed 3-manifolds with b₁=1

2021/08/30 by Johnson, Peter K.
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2108.13548

Abstract

In \citeMR1957829, Ozsváth and Szabó use Heegaard Floer homology to define numerical invariants d_1/2 and d_-1/2 for 3-manifolds Y with H₁(Y;ℤ)≅ ℤ. We define involutive Heegaard Floer theoretic versions of these invariants analogous to the involutive d invariants d and \underlined defined for rational homology spheres by Hendricks and Manolescu in \citeMR3649355 . We prove their invariance under spin integer homology cobordism and use them to establish spin filling constraints and 0-surgery obstructions analogous to results by Ozsváth and Szabó for their Heegaard Floer counterparts d_1/2 and d_-1/2. We then apply calculation techniques of Dai and Manolescu developed in \citeMR4021102 and Rustamov in \citeRustamov to compute the involutive Heegaard Floer homology of some negative semi-definite plumbed 3-manifolds with b₁ =1. By combining these calculations with the 0-surgery obstructions, we are able to produce an infinite family of small Seifert fibered spaces with weight 1 fundamental group and first homology ℤ which cannot be obtained by 0-surgery on a knot in S³, extending a result of Hedden, Kim, Mark, and Park in \citeMR4029676.

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