Searcharxiv⌕ Search

arXiv subjects

Christopher M. Herald

Publications and source records attributed to Christopher M. Herald.

11 recordsLinked to original sources
arXiv2026

A differential topology proof that the $SU(2)$ character variety of the genus two surface is homeomorphic to ${\mathbb C} P^3$

We provide a proof that the $SU(2)$ character variety of a genus two surface, $\chi(F_2)$, is a closed compact manifold, and a proof of the Narasimhan-Ramanan theorem that $\chi(F_2)$ is homeomorphic to ${\mathbb C} P^3$. This is done entirely in the language of $SU(2)$ representations, differential topology and elementary algebraic topology. It avoids the Narasimhan-Seshadri correspondence, clarifying the nature of Lagrangian immersions into $\chi(F_2)$ induced by 3-manifolds with genus two boundary. We give examples of such Lagrangian immersions and describe a correspondence from multicurves in the pillowcase to Lagrangian immersions in $\chi(F_2)$, induced by a 2-stranded tangle in a punctured genus 2 handlebody. We give an example of a non-transverse pair of smooth Lagrangians in $\chi(F_2)$ induced by a genus 2 Heegaard splitting of $(S^3,W)$ for the ``linked eyeglasses" web $W$, which are made transverse, and hence the corresponding Chern-Simons function Morse, using Goldman flows/holonomy perturbations along embedded curves in the Heegaard surface.

math.GT↗
arXiv2024

An endomorphism on immersed curves in the pillowcase

We examine the holonomy-perturbed traceless SU(2) character variety of the trivial four-stranded tangle {p_1,p_2,p_3,p_4} X [0,1] in S^2 X [0,1] equipped with a strong marking, either an earring or a bypass. Viewing these marked tangles as endomorphisms in the cobordism category from the four-punctured sphere to itself, we identify the images of these endomorphisms in the Weinstein symplectic partial category under the partially defined holonomy-perturbed traceless character variety functor. We express these endomorphisms on immersed curves in the pillowcase in terms of doubling and figure eight operations and prove they have the same image.

math.GT↗
arXiv2022

The correspondence induced on the pillowcase by the earring tangle

The earring tangle consists of four strands 4pt x [0,1] inside S^2 x [0,1] and one meridian around one of the strands. Equipping this tangle with a nontrivial SO(3) bundle, we show that its traceless SU(2) flat moduli space is topologically a smooth genus three surface. We also show that the restriction map from this surface to the traceless flat moduli space of the boundary of the earring tangle is a particular Lagrangian immersion into the product of two pillowcases. The latter computation suggests that figure eight bubbling -- a subtle degeneration phenomenon predicted by Bottman and Wehrheim -- appears in the context of traceless character varieties.

math.GT↗
arXiv2021

The Fukaya category of the pillowcase, traceless character varieties, and Khovanov Cohomology

For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We show that this twisted complex is a functorial invariant of the isotopy class of the tangle, and that it provides a factorization of Bar-Natan's functor from the tangle cobordism category to chain complexes. In particular, the hom set of our invariant with a particular non-compact Lagrangian associated to the trivial tangle is naturally isomorphic to the reduced Khovanov chain complex of the closure of the tangle. Our construction comes from the geometry of traceless SU(2) character varieties associated to resolutions of the tangle diagram, and was inspired by Kronheimer and Mrowka's singular instanton link homology.

math.GT↗
arXiv2015

Holonomy perturbations in a cylinder, and regularity for traceless SU(2) character varieties of tangles

The traceless $SU(2)$ character variety $R(S^2,\{a_i,b_i\}_{i=1}^n)$ of a $2n$-punctured 2-sphere is the symplectic reduction of a Hamiltonian $n$-torus action on the $SU(2)$ character variety of a closed surface of genus $n$. It is stratified with a finite singular stratum and a top smooth symplectic stratum of dimension $4n-6$. For generic holonomy perturbations $π$, the traceless $SU(2)$ character variety $R_π(Y,L)$ of an $n$-stranded tangle $L$ in a homology 3-ball $Y$ is stratified with a finite singular stratum and top stratum a smooth manifold. The restriction to $R(S^2, 2n)$ is a Lagrangian immersion which preserves the cone neighborhood structure near the singular stratum. For generic holonomy perturbations $π$, the variant $R_π^\natural(Y,L)$, obtained by taking the connected sum of $L$ with a Hopf link and considering $SO(3)$ representations with $w_2$ supported near the extra component, is a smooth compact manifold without boundary of dimension $2n-3$, which Lagrangian immerses into the smooth stratum of $R(S^2,\{a_i,b_i\}_{i=1}^n)$. The proofs of these assertions consist of stratified transversality arguments to eliminate non-generic strata in the character variety and to insure that the restriction map to the boundary character variety is also generic. The main tool introduced to establish abundance of holonomy perturbations is the use of holonomy perturbations along curves $C$ in a cylinder $F\times I$, where $F$ is a closed surface. When $C$ is obtained by pushing an embedded curve on $F$ into the cylinder, we prove that the corresponding holonomy perturbation induces one of Goldman's generalized Hamiltonian twist flows on the $SU(2)$ character variety $\mathcal{M}(F)$ associated to the curve $C$.

math.GT↗
arXiv2014

The pillowcase and traceless representations of knot groups II: a Lagrangian-Floer theory in the pillowcase

We define an elementary relatively $\mathbb Z/4$ graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram obtained by taking traceless $SU(2)$ character varieties of 2-tangle decompositions of knots. Calculations for torus knots are explained in terms of pictures in the punctured plane. The relation to the reduced instanton homology of knots is explored.

math.GT↗
arXiv2003

The Integer Valued SU(3) Casson Invariant for Brieskorn spheres

We develop techniques for computing the integer valued SU(3) Casson invariant. Our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the perturbed flat moduli space. These techniques, together with Bott-Morse theory and the splitting principle for spectral flow, are applied to calculate the invariant for all Brieskorn homology spheres.

math.GT↗
arXiv2000

An integer valued SU(3) Casson invariant

We define an integer valued invariant of homology spheres using the methods of SU(3) gauge theory and study its behavior under orientation reversal and connected sum.

math.GT↗
arXiv1998

The SU(3) Casson Invariant for Integral Homology 3-Spheres

We derive a gauge theoretic invariant of integral homology 3-spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, perturbed flat orbits. Our formula for the correction term generalizes that given by Walker in his extension of Casson's SU(2) invariant to rational homology 3-spheres.

math.DG↗