37 problems

String homology–Legendrian contact homology conjecture. For any compact oriented submanifold $K$ in $\mathbb{R}^n$, $H^{\mathrm{string}}_*(\mathbb{R}^n,K)$ is isomorphic to the Leg…

Let $M$ be a simply connected compact oriented smooth manifold, let $LM=\operatorname{map}(S^1,M)$ be its free loop space, and write $\mathbb{H}_*(LM)=H_{*+d}(LM)$ for shifted free…

Let $G$ be the compact Lie group and $EG$ the universal principal $G$-bundle, with $Ad(EG)$ its adjoint bundle and $T_v$ its vertical tangent bundle. Let $LBG^{-TBG}$ denote the as…

Legendre-transform equivalence conjecture. The natural map induced by the Legendre transform,

Floer-theoretic string topology operations conjecture. For every marked chord diagram $\Gamma$, there is an umkehr map

Geometric deformation conjecture. (a) For each oriented $3$-manifold, the epimorphism

Converse characterization conjecture. If $(\mathfrak{I}_{\hat\Psi},\mathfrak{t}_{\hat\Psi})$ is a commutative Frobenius algebra in

Let $M$ be a smooth closed manifold, let $\mathcal{H}(M)$ be its stable $h$-cobordism space, and suppose the higher Reidemeister trace constructions define a map … Let…

Let $E\to B$ be a smooth fiber bundle with fiber a smooth closed manifold $M$, and let $f:E\to M\times B$ be a fiberwise homotopy equivalence over $B$. Suppose there are spectral o…

Let $Q$ be a manifold and let $W$ be a Liouville domain. Write $\widehat{\mathbb{H}}_*^{S^1}\Lambda$ for Rabinowitz string homology, $S\mathbb{H}_*^{S^1}({\partial}W)$ for $S^1$-eq…

Let $Q$ be a manifold, let $W$ be a Liouville domain of dimension $2n$, and let $\widehat{\mathbb{H}}_*\Lambda$, $\overline{\mathbb{H}}_*\Lambda$, $S\mathbb{H}_*({\partial}W)$, and…

Let $\mathbfcal M_{g,k}$ denote the logarithmic stack of formal curves with framings, and let $\textsf N^{fr}_{\mathrm{LMM}}$, $\textsf N^{fr}_{\mathrm{MHS}}$, and…

Let $d$ be an integer, and let $\textsf{OGC}_{2d+1}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ be the oriented graph complex and…

Let $M$ be a closed oriented C^-manifold, let $\Omega(M)$ denote its de Rham algebra, and let $\Omega(M)^\vee[-\dim M]$ be the shifted dual complex. Write…

Let $M$ be an oriented closed manifold of dimension $n$. Let $C^{\mathrm{sing}}(\Loop_{\Sph{1}}M;\R)$ be the smooth singular chain complex with boundary $\Bdd$, and let…

Let $M$ be a closed oriented manifold of dimension $n$, and let \H_{\mathrm{dR}} be its de Rham cohomology. Let \DBCyc\,\H_{\mathrm{dR}}[2-n] denote a suitable shifted cyclic c…

Let $M$ be a connected, oriented, closed Riemannian manifold with $\HDR^1(M)=0$, and suppose that $M$ is formal. In the situation of the equivalence conjecture, let $\PMC$ be the C…

Let $M$ be a connected, oriented, closed Riemannian $n$-manifold with $\HDR^1(M)=0$. Let $\DR(M)$ be its de Rham complex, $\HDR(M)$ its de Rham cohomology, and…

Let $V$ be a differential Poincaré duality algebra with \H^0(V)=\operatorname{Span}\{1\} and \H^1(V)=0, and let $\Model_1$ and $\Model_2$ be Poincaré duality models satisfying…

Let $(Q,q_0)$ and $(Q',q'_0)$ be compact pointed Riemannian manifolds. For each $t\in\mathbf{R}$, let $F_{<t}C_*(\Omega_{q_0}Q;\mathbf{C})$ denote the subcomplex generated by loops…

Let $Y$ be the unit cotangent bundle of a hyperbolic surface, and let $(M=\Sigma \times Y,\xi_k)$ be the corresponding $5$-dimensional contact manifold from Corollary … have untwis…

Let $G$ be a connected compact Lie group of dimension $d$, let $BG$ be its classifying space, and let $LBG$ denote the free loop space of $BG$. Write $S_*(G)$ for the singular chai…

Orientation-cover triviality conjecture. This double cover is trivial. The source does not provide a proof or status evidence for this assertion.

Let $\mathcal{SD}^{\mathrm{u}}$ be the space of unoriented string diagrams, let $\mathcal{SD}^{\mathrm{u}}/\sim$ be its quotient by the stated equivalence relation, and let…

Let $\mathcal{SD}^{\mathrm{u}}$ be the space of unoriented string diagrams, and let $\mathcal{M}$ be the moduli space of Riemann surfaces with parameterized boundary. Unoriented st…