Talk

This talk is the second part of the series on spectral sequences from Khovanov homology to Heegaard–Floer homology.
We focus on Szabó’s geometric spectral sequence over Z2, whose infinity page is conjectured to coincide with the
Heegaard–Floer homology of the branched double cover of the mirror of a link, and we discuss Beier’s lift of this spectral
sequence to integer coefficients using odd Khovanov homology.

One way of studying compact manifolds is either by endowing them with a geometry, following the approach used by Thurston for
3-manifolds, or by endowing their fundamental group with a (large scale) geometry. I will explain one particular way of defining
nonpositive curvature for such a fundamental group, using bicombings (an idea introduced by Busemann and pursued by Thurston)
and the impact that this property has on the (coarse) topology of the manifold itself and of its universal cover.

There exists an equivalence of (∞,2)-categories between the (∞,2)-category of complexes of stable ∞-categories and that of 2-simplicial stable ∞-categories, established by Dyckerhoff, which categorifies the classical Dold–Kan correspondence. It is well known that every simplicial abelian group is in particular a Kan complex, i.e. it admits certain horn fillers.

A one-parameter family of self-adjoint Fredholm operators has a well-known integer-valued invariant, the spectral flow. It counts (with signs) the number of eigenvalues changing their sign along the way. For loops of elliptic operators on a closed manifold, the spectral flow was computed by Atiyah, Patodi, and Singer (1976) in terms of topological data of a loop. But if a manifold has non-empty boundary, then boundary conditions come into play, and situation becomes more complicated.