arXiv subjects

Adela YiYu Zhang

Publications and source records attributed to Adela YiYu Zhang.

6 recordsLinked to original sources

arXiv2025

Open 2D TFTs admit initial open-closed extensions

We show that any open 2-dimensional topological field theory valued in a symmetric monoidal $\infty$-category (with suitable colimits) extends canonically to an open-closed field theory whose value at the circle is the Hochschild homology object of its value at the disk. As a corollary, we obtain an action of the moduli spaces of surfaces on the Hochschild homology object of $E_1$-Calabi-Yau algebras. This provides a space level refinement of previous work of Costello over $\mathbb{Q}$ and Wahl-Westerland and Wahl over $\mathbb{Z}$, and serves as a crucial ingredient to Lurie's "non-compact cobordism hypothesis" in dimension 2. As part of the proof we also give a description of slice categories of the d-dimensional bordism category with boundary, which may be of independent interest.

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arXiv2025

Inertia groups of $(n-1)$-connected $2n$-manifolds

In this paper, we compute the inertia groups of $(n-1)$-connected, smooth, closed, oriented $2n$-manifolds where $n \geq 3$. As a consequence, we complete the diffeomorphism classification of such manifolds, finishing a program initiated by Wall sixty years ago, with the exception of the $126$-dimensional case of the Kervaire invariant one problem. In particular, we find that the inertia group always vanishes for $n \neq 4,8,9$ -- for $n \gg 0$, this was known by the work of several previous authors, including Wall, Stolz, and Burklund and Hahn with the first named author. When $n = 4,8,9$, we apply Kreck's modified surgery and a special case of Crowley's $Q$-form conjecture, proven by Nagy, to compute the inertia groups of these manifolds. In the cases $n=4,8$, our results recover unpublished work of Crowley--Nagy and Crowley--Olbermann. In contrast, we show that the homotopy and concordance inertia groups of $(n-1)$-connected, smooth, closed, oriented $2n$-manifolds with $n \geq 3$ always vanish.

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arXiv2025

Quillen homology of spectral Lie algebras with application to mod $p$ homology of labeled configuration spaces

We provide a general method computing the mod $p$ Quillen homology of algebras over a monad that parametrizes the structure of mod $p$ homology of spectral Lie algebras. This is the $E^2$-page of the bar spectral sequence converging to the mod $p$ topological Quillen homology of spectral Lie algebras. The computation of the Quillen homology of the trivial algebra allows us to deduce that the $\mathbb F_p$-linear spectral Lie operad is not formal. As an application, we study the mod $p$ homology of the labeled configuration space $B_k(M;X)$ of $k$ points in a manifold $M$ with labels in a spectrum $X$, which is the mod $p$ topological Quillen homology of a certain spectral Lie algebra by a result of Knudsen. We obtain general upper bounds for the mod $p$ homology of $B_k(M;X)$, as well as explicit computations for small $k$. When $p$ is odd, we observe that the mod $p$ homology of $B_k(M^n;S^r)$ for small $k$ depends on and only on the cohomology ring of the one-point compactification of $M$ when $n+r$ is even. This supplements and contrasts with the result of Bödigheimer-Cohen-Taylor when $n+r$ is odd.

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arXiv2025

Operations on spectral partition Lie algebras and TAQ cohomology

We determine all natural operations and their relations on the homotopy groups of spectral partition Lie algebras, which coincide with $\mathbb{F}_p$-linear topological André-Quillen cohomology operations at any prime. We construct unary operations and a shifted restricted Lie algebra structure on the homotopy groups of spectral partition Lie algebras. Then we prove a composition law for the unary operations, as well as a compatibility condition between unary operations and the shifted Lie bracket with restriction up to a unit for the restriction. Comparing with Brantner-Mathew's result on the ranks of the homotopy groups of free spectral partition Lie algebras, we deduce that these generate all natural operations, thereby also recovering unpublished results of Kriz and Basterra-Mandell on $\mathbb{F}_p$-linear TAQ cohomology operations. As a corollary, we determine the structure of natural operations on mod $p$ $\mathbb{S}$-linear TAQ cohomology.

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arXiv2024

Bounding the $K(p-1)$-local exotic Picard group at $p>3$

In this paper, we bound the descent filtration of the exotic Picard group $κ_n$, for a prime number p>3 and n=p-1. Our method involves a detailed comparison of the Picard spectral sequence, the homotopy fixed point spectral sequence, and an auxiliary $β$-inverted homotopy fixed point spectral sequence whose input is the Farrell-Tate cohomology of the Morava stabilizer group. Along the way, we deduce that the K(n)-local Adams-Novikov spectral sequence for the sphere has a horizontal vanishing line at $3n^2+1$ on the $E_{2n^2+2}$-page. The same analysis also allows us to express the exotic Picard group of $K(n)$-local modules over the homotopy fixed points spectrum $\mathrm{E}_n^{hN}$, where N is the normalizer in $\mathbb{G}_n$ of a finite cyclic subgroup of order p, as a subquotient of a single continuous cohomology group $H^{2n+1}(N,π_{2n}\mathrm{E}_n)$.

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arXiv2022

Mod $p$ homology of unordered configuration spaces of surfaces

We provide a short proof that the dimensions of the mod $p$ homology groups of the unordered configuration space $B_k(T)$ of $k$ points in a torus are the same as its Betti numbers for $p>2$ and $k\leq p$. Hence the integral homology has no $p$-power torsion. The same argument works for the punctured genus $g$ surface with $g>0$, thereby recovering a result of Brantner-Hahn-Knudsen via Lubin-Tate theory.

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