We study 4-dimensional homology cobordisms without 3-handles via Heegaard and instanton Floer homologies, character varieties, and Thurston geometries. We provide obstructions to such cobordisms arising from each of these theories, and illustrate some topological applications.

We re-derive Manolescu's unoriented skein exact triangle for knot Floer homology over F 2 combinatorially using grid diagrams, and extend it to the case with Z coefficients by sign refinements. Iteration of the triangle gives a cube of resolutions that converges to the knot Floer homology of an oriented link. Finally, we re-establish the homological σ-thinness of quasi-alternating links.

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