arXiv subjects

Ödül Tetik

Publications and source records attributed to Ödül Tetik.

3 recordsLinked to original sources

arXiv2026

Topological Obstructions to Riemannian-Lorentzian Continuation

Motivated by problems in instanton continuation, we establish geometro-topological selection rules for continuing a Riemannian region into a Lorentzian region through a smooth transverse change of signature. Given a collar-gluing $M=\overline{M_R} \cup_{\mathcal{H}}\overline{M_L}$ of a smooth manifold $M$ along a hypersurface $\mathcal{H}$, we show that an admissible signature-changing metric of the form $\widetilde g=g-fV^\flat\otimes V^\flat$ with $g$ a Riemannian metric, having Riemannian signature on $M_R$ and Lorentzian signature on $M_L$, exists if and only if a relative Euler class obstruction on $\overline{M_L}$ vanishes. This obstruction combines ordinary Euler characteristics $χ$ in a manner that depends on which components of $\mathcal{H}$ represent big bangs and which represent big crunches. It also restricts the choice of $M_R$; for example, if $M$ is closed and $\dim M$ is even, then it requires$χ(\overline{M_R})=χ(M)$. We also determine and classify the possible compactness types of $\overline{M_R}$, $\mathcal{H}$, and $\overline{M_L}$ occurring in such a collar-gluing construction, and construct explicit signature-changing metrics in each case. Topology change can be entirely contained within the Riemannian region when $M$ is the interior of a compact manifold with boundary.

math.DG↗

arXiv2025

The stratified Grassmannian and its depth-one subcategories

We introduce a tangential theory for linked smooth manifolds of depth $1$, i.e., for spans $\mathfrak{S}=(M\oversetπ{\twoheadleftarrow} L\oversetι{\hookrightarrow}N)$ of smooth manifolds where $π$ is a fibre bundle and $ι$ is a closed embedding. The tangent classifier of $\mathfrak{S}$ is given as a topological span map $\mathfrak{S}\to B\mathrm{O}(n,m)$ where $B\mathrm{O}(n,m)=(B\mathrm{O}(n)\twoheadleftarrow B\mathrm{O}(n)\times B\mathrm{O}(m)\hookrightarrow B\mathrm{O}(n+m))$. We show that this recovers and generalises the tangential theory introduced by Ayala, Francis and Rozenblyum for conically smooth stratified spaces by constructing fully faithful functors $\mathbf{EX}(B\mathrm{O}(n,m))\hookrightarrow\mathbf{V}^{\hookrightarrow}$ of quasi-categories, where $\mathbf{EX}$, introduced in a prequel, takes the exit path quasi-category of the span, and $\mathbf{V}^{\hookrightarrow}$ is a quasi-category model of the infinite stratified Grassmannian of AFR. This result has analogues for other classical structure groups and for Stiefel manifolds. We thus reduce the classification of conically smooth bundles over depth-$1$ posets to that of ordinary bundles on linked smooth manifolds.

math.AT↗

arXiv2025

Linked spaces and exit paths

Conically smooth spaces (CSSs), introduced by Ayala, Francis and Tanaka, constitute a large class of singular spaces including Whitney-stratified spaces. We reduce the stratified topology of CSSs over depth-$1$ posets to the ordinary topology of linked smooth manifolds, i.e., spans $M\xleftarrowπ L\xrightarrowιN$ of smooth manifolds where $π$ is a fibre bundle and $ι$ is a closed embedding. To that end, we introduce explicit exit path quasi-categories (EPCs) for linked spaces and prove that this induces a fully faithful functor from a quasi-category of linked spaces to the quasi-category of all quasi-categories whose essential image includes the Lurie--MacPherson EPCs of CSSs over depth-$1$ posets. We use linked smooth manifolds to resolve various weaker versions of a conjecture of Ayala--Francis--Rozenblyum in the negative by exhibiting quasi-categories with conservative functors to $\{0<1\}$ satisfying certain finiteness conditions but which are not equivalent to EPCs of CSSs. In a sequel, we develop a tangential theory for linked smooth manifolds and reduce the classification conically smooth bundles over depth-$1$ posets to that of ordinary bundles on linked smooth manifolds.

math.AT↗