McDuff and Segal proved homological stability for unordered configuration spaces of connected manifolds with non-empty boundary. Later, Church proved representation stability for ordered configuration spaces on compact manifolds. These stability patterns extend to surface braid groups and pure surface braid groups, respectively, whenever the associated configuration space is an Eilenberg--Maclane space. However, these results do not apply to braid groups and pure braid groups on $S^2$ and $\RP^2$ since ordered and unordered configuration spaces on these surfaces are not Eilenberg--Maclane spaces. This paper will use results by Cohen--Pakianathan and the theory of $\FI$-homology to extend known stability patterns to these unknown cases. This is a special case of a more general result proven in this paper: the equivariant homology of ordered and unordered configuration spaces exhibit representation stability and stable periodicity respectively.
We study representations of compact oriented surface groups into $\mathrm U(p)$ with elliptic-unipotent boundary holonomies, meaning that no boundary holonomy has eigenvalue $1$. We prove that the signature of the associated flat Hermitian bundle completely determines the connected component, and that every component i...
We solve the three-dimensional symplectization question for several examples. These include small Seifert fibered $L$-spaces, a family of Whitehead-link surgery $L$-spaces, $S^1\times S^2$, certain surgeries on the figure-eight knot, explicit families of Brieskorn spheres, and arbitrary finite connected sums mixing the...
This paper establishes necessary and sufficient conditions for locally flat knotted surfaces in simply-connected $4$-manifolds to be equivalent. For surfaces with knot group $\mathbb{Z}_d$, we extend results of Lee-Wilczynski from spheres to surfaces of arbitrary genus; the surfaces are permitted to be nonorientable an...
We study the monodromy action of the mixed braid group $B_{n,\mathcal{P}}$ on the first cohomology of cyclic branched covers of $\mathbb{P}^1$, which are mutually determined by a partition of branch points by equal ramification. The monodromy representation splits into irreducible representations on the $t$-eigenspaces...
We consider objects in the Fukaya category of a class of K3 surfaces defined by certain Lagrangian spheres. Assuming that homological mirror symmetry for K3 surfaces holds with sufficiently strong (expected) properties, we prove that, in this case, stability with respect to a suitable Bridgeland stability condition imp...
For the unordered discrete configuration space $\mathrm{UD}_n(\mathsf{\Gamma})$ of $n$ particles on a connected finite graph $\mathsf{\Gamma}$, we construct an explicit factor system on its universal cover. Its factors are encoded by legal pairs, namely subgraphs equipped with particle distributions. The nesting, ortho...
B. An, Sang-Rok Oh, JiHoon Park· 0 citations
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