João Lobo Fernandes | Searcharxiv

arXiv2026

Stable moduli spaces of odd-dimensional manifold triads

We establish a homotopy-theoretic description of the homology of stable moduli spaces of $(2n+1)$-dimensional manifold triads $(N, \partial^h N, \partial^v N)$ with fixed $\partial^v N$, whenever $n \geq 3$ and $(N, \partial^h N)$ is $1$-connected. Stabilization is performed by taking boundary connected sum with $S^n \times D^{n+1}$. This is an analog of earlier work of Galatius and Randal-Williams for even-dimensional manifolds with fixed boundary, and it extends a previous result by Botvinnik and Perlmutter. As a byproduct, we obtain an analog for odd-dimensional triads of Kreck's stable diffeomorphism classification of even-dimensional manifolds.

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arXiv2026

Diffeomorphism groups of solid tori and the rational pseudoisotopy stable range

We compute the rational homotopy groups of the classifying space $\mathrm{BDiff}_{\partial}(S^1 \times D^{d-1})$ of the topological group of diffeomorphisms of $S^1 \times D^{d-1}$ fixing the boundary for $d \geq 6$, in a range of degrees up until around $d$. This extends results of Budney-Gabai, Bustamante-Randal-Williams, and Watanabe. As consequences of this computation, we determine the rational pseudoisotopy stable range for compact spin manifolds with fundamental group $\mathbf{Z}$ of dimension $d\geq 6$ to be $[0,d-5]$, and compute in this range the rational homotopy groups of $\mathrm{BDiff}_{\partial}(S^1 \times N)$ for compact simply-connected spin $(d-1)$-manifolds $N$. Finally, by combining our results with work of Krannich-Randal-Williams and Kupers-Randal-Williams on $\mathrm{BDiff}_{\partial}(D^d)$, we compute the rational homotopy groups of the space $\mathrm{Emb}_{\partial}(D^{d-2}, D^d)$ of long knots in codimension 2 for $d \geq 6$, again in the same range.

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