17 problems

Order-two bordism conjecture. The group $\pi_{11}M\mathfrak m_c$ is cyclic of order $2$, the bordism class of $(N,0)$ represents its generator, and the mod-$2$ index of the…

Let $H(m,n)$ denote a Milnor manifold and $P(r,s)$ a Dold manifold. The Milnor manifold $H(m,n)$ is non-bounding if it does not bound, and it is among those not considered in Propo…

Pull-back conjecture. The square

Let $Q_8$ be the quaternion group, let $C_2$ be its indicated subgroup, and let $N_{C_2}^{Q_8} MU\mathbb{R}$ denote the quaternionic norm of real bordism. For a subgroup $L$ of…

Functoriality conjecture. The class $\Omega_!(g\circ f)$ should be identified, up to a canonical bordism, with

Let $C_2$ be the cyclic group of order two, let $\operatorname{MSpin}^c_{\operatorname{\mathbb{R}}}$ denote Real spin bordism, let $\mathcal{P}$ be the indexing set for the Anderso…

Let $(\xi, \mathcal P)$ be a pair consisting of a tangential structure and a characteristic class as in the definition of a characteristic pair, and let $\xi'$ be a tangential stru…

Fix an integer $n$. Let $\Flow_{\tau_{\leq n}}$ be the category of flow categories with morphisms given by homotopy classes of $\tau_{\leq n}$-morphisms, where $\tau_{\leq n}$ deno…

Let $A=\mathbb R\times[0,1]$. Denote by $\mathbf{OC}^{\mathsf{imm}}_{\mathsf{moderate \leq 2}}(A)$ the abelian group of moderate immersed open-curve doodle classes, and by…

Let $A=\mathbb R\times[0,1]$. Denote by $\mathbf B^{\mathsf{imm}}_{\mathsf{moderate \leq 2}}(A)$ the abelian group of moderate immersed doodle bordism classes, and by $\mathbb Z_2$…

Let $c9$ be a tangential structure, and suppose a consistent $n$-dimensional theory of quantum gravity sums over spacetime backgrounds carrying a $c9$-structure. The associated bor…

Let $G=\{G_d,s_d,\rho_d\}_{d\in\mathbb{Z}_{\geq 0}}$ be a tangential structure, and let $I\mathbb{C}^{\times}$ denote the Brown–Comenetz dual of the sphere. Write $I\Omega^G$ for t…

Let $BPGL\langle m\rangle$ be the truncated motivic Brown–Peterson spectrum, and let its homological slice spectral sequence (HSSS) have differentials $d_r$. The differentials disp…

Let $f$ and $g$ be $n$-morphisms in the oriented bordism category $\mathbf{Bord}^{\mathrm{or}}_n$. Completeness conjecture. If $f$ and $g$ are not equivalent, then there exists a s…

Let $V$ be a nonsingular real algebraic variety and let $M$ be a compact smooth submanifold of $V$. Suppose that the unoriented bordism class of the inclusion map…

Let $C_{2^n}$ be the cyclic group of order $2^n$, and let $a_{\sigma_n}$ and $u_{\rho_n-\sigma_n}$ denote the Euler and orientation classes associated to the representations…

Bordism invariance conjecture. The bordism $\hat{W}$ between $\hat{\Sigma}_l$ and $\hat{\Sigma}_l'$ gives a natural isomorphism