81 problems

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 closed oriented manifold, and let $\StringH(\Loop M)$ carry the Chas–Sullivan string bracket $\StringOp_2$ and cobracket $\StringCoOp_2$. Let $\OPQ_{210}^\PMC$ and…

Cohen–Jones BV algebra conjecture. The identification $F$ is an isomorphism of BV algebras.

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…

Free-loop-space operations conjecture. These operations should extend to operations

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

Let $R$ be a based (that is, augmented) algebra over $C_{\ast}\mathcal{C}_k$, where $k\geq 1$, and let its Quillen and Hochschild cohomologies be taken in the category of based…

Let $\mathbb P^n$ be complex projective space, let $L^2\mathbb P^n$ denote its double loop space, and let $\hol{}(n)$ and $\rat{}(n)$ denote the holomorphic and rational mapping sp…

Let $\mathcal CF^{\mu}_{p,q}$ be the space of homological string topology operations associated with surfaces of genus $g$, with $p$ incoming and $q$ outgoing boundary components.…

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

Cohen's conjecture. The quantum field theory of Chas–Sullivan on $M$ is isomorphic to the Floer quantum field theory of $T^*M$.

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

Locality conjecture. The full package of chain-level string topology, in particular the $\infty$-involutive Lie bialgebra (or $IBL_{\infty}$-algebra) structure on the $S^1$-equivar…

O's conjecture. When $Q=\mathbb{R}^n$, the graded algebra $H^{\mathrm{string}}_*(\mathbb{R}^n,K)$ is isomorphic to the Legendrian contact homology of $\Lambda_K$ with coefficients…

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…

Rivera–Takeda–Wang's conjecture. For an appropriate choice of the lift of $\bbDelta(1)$, the secondary coproduct coincides with the loop coproduct; in particular, the appropriate c…

String topology coproduct conjecture. The string topology coproduct of $M$ is trivial.

Let $X$ be an oriented smooth manifold. Let $(\operatorname{coHoch}_\bullet(\mathcal{C}(X)),d,P)$ be the associated mixed complex, and consider its relative cyclic chains. An…

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 $C_*^{S^1}$ be the $S^1$-equivariant chain complex used in the source, with chain-level bracket $\{\cdot,\cdot\}_{\mathit{str}}$ defined from the erasing map, marking map, 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…