arXiv subjects

Jiaxi Zha

Publications and source records attributed to Jiaxi Zha.

5 recordsLinked to original sources

arXiv2026

On The Rational Realization of Even-dimensional Spheres and Products of Eilenberg--MacLane Spaces as Classifying Spaces

In this paper, we study the rational realization problem for the classifying space $\B(X)$. We prove that if $S^{2n}$ is realized as $\B(X)$ for a simply-connected space $X$, then $X$ is $π$-infinite and has vanishing rational Gottlieb elements above degree $2n-1$. In particular, even-dimensional spheres cannot be realized as $B\mathrm{aut}_1(X)$ for any simply-connected $π$-finite space $X$. We also prove that, for all $n\geq 2$ and $s,t\geq 1$, the product of Eilenberg--MacLane spaces $K(\Q^s,n)\times K(\Q^t,n+1)$ cannot be realized as $B\mathrm{aut}_1(X)$ for any simply-connected $π$-finite space $X$. Moreover, we prove that if $r\geq 2$ and $n\geq 3$ and $K(\Q^r,n)$ is realized as $\B(X)$ for a $π$-finite space $X$, then $X\simeq_{\Q}K(\Q^r,n-1)$. The proofs are based on two structural results for Gottlieb elements in the derivation Lie algebra of a Sullivan minimal model, which provide a uniform method for these realization problems.

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arXiv2026

Logarithmic topological Hochschild homology of the truncated Brown--Peterson spectrum

We study logarithmic topological Hochschild homology (log THH) of the truncated Brown--Peterson spectra $\mathrm{BP}\langle n\rangle$. We first construct compatible prelog structures on $\mathrm{BP}\langle n\rangle$ generated by the classes $v_i$. Then we compute the $V(n)$-homotopy groups of log THH of $\mathrm{BP}\langle n\rangle$ for the prelog structures generated by $v_n$ and $(v_1,\cdots,v_n)$, under the assumption that the Smith--Toda complex $V(n)$ exists as a ring spectrum. At height $2$ and for $p\geq7$, we also compute the $V(2)$-homotopy groups of log THH of $\mathrm{BP}\langle 2\rangle/v_1$ relative to $v_2$.

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arXiv2026

Topological CoHochschild Homology and Thom Spectra

For an $\E$-ring spectrum $R$ and a map $f:X\to Pic(R)$ of spaces, the Thom spectrum $\T f$ is a comodule over $R\otimes\Si X$. In this paper we study the topological coHochschild homology of $R\otimes\Si X$ with coefficient $\T f$. More concretely, for a simply connected space $X$, we will give a filtration on $\mathrm{coTHH}^R(\T f;R\otimes\Si X)$ via the cellular structure of $X$. Furthermore, we will reduce the computation of $\mathrm{coTHH}^R(\T f;R\otimes\Si X)$ to that of $\mathrm{coTHH}^R(R\otimes\Si G)$ for some group-like $\mathbb{E}_1$-spaces. Finally, we will use these results to study properties of $\mathrm{coTHH}^R(\T f;R\otimes\Si X)$.

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arXiv2025

On topological coHochschild homology and cotensor products

In this work, we first study the cotensor product of comodules in the $\infty$-category $\mathrm{Mod}_R$ for a connected $\mathbb{E}_{\infty}$-ring spectrum $R$. We then apply these results to analyze higher coalgebra structures of topological coHochschild homology (coTHH) and establish its Morita-Takeuchi invariance, which are precisely dual to the corresponding properties of topological Hochschild homology.

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arXiv2025

Cohomology of $BPU_n$ and rings of invariants of Weyl groups

Let $PU_n$ denote the projective unitary group of rank $n$ and $BPU_n$ be its classifying space, for $n>1$. Using the Serre spectral sequence associated to the fibration $BU_n\to BPU_n\to K(\mathbb{Z},3)$, we compute the integral cohomology group of $BPU_n$ in dimensions $\leq 14$. In addition, we determine the ring structure of $H^*(BPU_n;\mathbb{Z})$ up to dimension $13$ by computing the ring of invariants $H^*(BT_{PU_n})^W$ of the Weyl group action in dimensions $\leq 12$.

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