We prove the strong Suslin reciprocity law conjectured by A. Goncharov. The Suslin reciprocity law is a generalization of the Weil reciprocity law to higher Milnor $K$-theory. The Milnor $K$-groups can be identified with the top cohomology groups of the polylogarithmic motivic complexes; Goncharov's conjecture predicts the existence of a contracting homotopy underlying Suslin reciprocity. The main ingredient of the proof is a homotopy invariance theorem for the cohomology of the polylogarithmic motivic complexes in the ‘next to Milnor’ degree. We apply these results to the theory of scissors congruences of hyperbolic polytopes. For every triple of rational functions on a compact projective curve over $\mathbb {C}$ we construct a hyperbolic polytope (defined up to scissors congruence). The hyperbolic volume and the Dehn invariant of this polytope can be computed directly from the triple of rational functions on the curve.

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over ${\mathbf {C}}$ there are $27$ lines, and over ${\mathbf {R}}$ the number of hyperbolic lines minus the number of elliptic lines is $3$. In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $\alpha$ in $L^*/(L^*)^2$. There is an equality in the Grothendieck–Witt group $\operatorname {GW}(k)$ of $k$,

\[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \]

$\operatorname {Tr}_{L/k}$

$\operatorname {GW}(L) \to \operatorname {GW}(k)$

${\mathbf {C}}$

${\mathbf {R}}$

$\mathbf {A}^1$

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq ~2$ we associate to an annular link $L$ a naive $\mathbb {Z}/r\mathbb {Z}$-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb {Z}[\mathbb {Z}/r\mathbb {Z}]$. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.

We show a rigidity theorem for the Seiberg–Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These non-smoothable topological families provide new examples of $4$-manifolds $M$ for which the inclusion maps $\operatorname {Diff}(M) \hookrightarrow \operatorname {Homeo}(M)$ are not weak homotopy equivalences. We shall also give a new series of non-smoothable topological actions on some spin $4$-manifolds.

We study a class of two-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the two-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $\pi$ is greater than the amplitude of the volume function. We also prove a K-theoretic criterion for a polynomial to be a factor of an $A$-polynomial and give a topological interpretation of its Mahler measure.

We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration. We also prove, along the way, a result of independent interest, giving sufficient conditions for smoothness of the total space of the relative compactified Jacobian of a family of locally planar curves.