arXiv subjects

Andreas Stavrou

Publications and source records attributed to Andreas Stavrou.

7 recordsLinked to original sources

arXiv2025

Surface configuration kernels

Let $\Sigma_{g,*}$ be a once-punctured oriented surface of genus $g$. We study the action of the mapping class group $\Gamma_{g,*}$ on the $n^{th}$ rational cohomology of the configuration space $\text{Conf}_n(\Sigma_{g,*})$ of injections $\{1,\ldots, n\}\hookrightarrow \Sigma_{g,*}$, and compare the kernel $J_{g,*}^{cfg}(n)$ of this action with the $n^{th}$ Johnson subgroup $J_{g,*}(n)$. We find high-rank abelian subgroups in the quotient $J_{g,*}^{cfg}(n)/J_{g,*}(n)$ arising from the higher Johnson images and from symplectic representation theory. In particular we refute a conjecture due to Bianchi--Miller--Wilson.

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arXiv2025

The mapping class group invariants of the truncated group ring

We compute the invariant subspace of the rational group ring of a surface, truncated by powers of the augmentation ideal, under the action of the mapping class group. The surface is compact, oriented with one boundary component. This provides the first group cohomology computation for the mapping class group with non-symplectic coefficients since Kawazumi-Souli\'e. Our computation is valid in a range growing with the genus.

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arXiv2025

The configuration functor of a punctured space

Let $U$ be a space whose one point compactification $U^*$ is a CW-complex for which the added point $*$ is the only $0$-cell. We observe that the configuration space $Conf_n(U)$ of $n$ numbered distinct points in $U$ has no closed support homology in degree $<n$ and prove that Borel-Moore homology group $H^{cl}_n(Conf_n(U))$ depends only on the fundamental group $\pi_1(U^*,*)$. We describe this homology group in terms of a presentation of $\pi_1(U^*,*)$. A case of interest is when $U$ is a connected closed oriented surface of positive genus minus a finite nonempty set. Then the mapping class group $Mod(U)$ of $U$ acts on both $\pi_1(U^*,*)$ and ${H^k}(Conf_n(U){)}\cong H^{cl}_ {2n-k}(Conf_n(U))$ and we prove that its action on the latter is through its action on the nilpotent quotient $\pi_1(U^*,*)/ \pi_1(U^*,*)^{(k+1)}$. Furthermore, we give an example of a mapping class of a once punctured closed surface $U$ which acts trivially on ${H^n}(Conf_n(U))$, but not on the nilpotent quotient $\pi_1(U^*,*)/ \pi_1(U^*,*)^{(n+1)}$. The former generalizes a theorem of Bianchi-Miller-Wilson and the latter disproves a conjecture of theirs.

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arXiv2024

Superposition of configurations and scanning

We endow the cohomology of configuration spaces of a manifold with a product arising from superposing configurations. We prove that, under the scanning isomorphism, this product corresponds to the cup-product of the section space of the standard scanning bundle of the manifold.

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arXiv2023

Homology of configuration spaces of surfaces modulo an odd prime

For a compact orientable surface $\Sigma_{g,1}$ of genus $g$ with one boundary component and for an odd prime number $p$, we study the homology of the unordered configuration spaces $C_\bullet(\Sigma_{g,1}):=\coprod_{n\ge0}C_n(\Sigma_{g,1})$ with coefficients in $\mathbb{F}_p$. We describe $H_*(C_\bullet(\Sigma_{g,1});\mathbb{F}_p)$ as a bigraded module over the Pontryagin ring $H_*(C_\bullet(D);\mathbb{F}_p)$, where $D$ is a disc, and compute in particular the bigraded dimension over $\mathbb{F}_p$. We also consider the action of the mapping class group $\Gamma_{g,1}$, and prove that the mod-$p$ Johnson kernel $\mathcal{K}_{g,1}(p)\subseteq\Gamma_{g,1}$ is the kernel of the action on $H_*(C_\bullet(\Sigma_{g,1};\mathbb{F}_p))$.

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arXiv2022

Non-trivial action of the Johnson filtration on the homology of configuration spaces

We let the mapping class group $Γ_{g,1}$ of a genus $g$ surface $Σ_{g,1}$ with one boundary component act on the homology $H_*(F_{n}(Σ_{g,1});\mathbb{Q})$ of the $n^{th}$ ordered configuration space of the surface. We prove that the action is non-trivial when restricted to the $(n-1)^{st}$ stage of the Johnson filtration, for all $n\ge 1$ and $g\ge 2$. We deduce an analogous result for closed surfaces.

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arXiv2021

Cohomology of configuration spaces of surfaces as mapping class group representations

We express the rational cohomology of the unordered configuration space of a compact oriented manifold as a representation of its mapping class group in terms of a weight-decomposition of the rational cohomology of the mapping space from the manifold to a sphere. We apply this to the case of a compact oriented surface with one boundary component and explicitly compute the rational cohomology of its unordered configuration space as a representation of its mapping class group. In particular, this representation is not symplectic, but has trivial action of the second Johnson filtration subgroup of the mapping class group.

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