2021-05-18

Kristen Hendricks, Surgery formulas for involutive Heegaard Floer homology

Tags: #projects/notes/seminars #geomtop #geomtop/Floer-theory Refs: Heegard-Floer homology

Reference: Kristen Hendricks, Surgery formulas for involutive Heegaard Floer homology. Stanford Topology Seminar.

• Want to study homology cobordism groups of 3-manifolds $$\Theta_{\mathbf{Z}}^3$$.

• We don’t understand the torsion in this group.
• Reduce to study of a group of “$$\iota$$ complexes”.
• Theorem: it has a $${\mathbf{Z}}^{\infty}$$ summand.
• See Seifert fibered spaces.

• People like HF because there are a lot of computational tools! In particular, a surgery formula.

22:01

Check out ideal sheaves, Birdgeland stability conditions.

• “Rotate” an exact sequence to get an exact triangle :

• Interpretation of the triangle: $$F = E \ominus {\mathcal{O}}(-n) = E \oplus {\mathcal{O}}(-n)[1]$$.
• Interpret $${\mathcal{O}}(-n)[1] = -{\mathcal{O}}(n)$$.
• What is an etale algebra?