40 problems

Let $G=\{G_d,s_d,\rho_d\}_{d\in\mathbb Z_{\ge 0}}$ be a system of compact Lie groups with structure maps $s_d\colon G_d\longrightarrow G_{d+1}$ and representations…

A closed manifold $M$ is almost flat if, for every $\epsilon>0$, it admits a Riemannian metric $g_\epsilon$ satisfying … where $K_{g_\epsilon}$ denotes sectional curvature. Farrell…

Swampland Cobordism Conjecture. The bordism groups of quantum gravity are trivial:

Let $M$ be a closed manifold and let $M\to E\xrightarrow{\pi}B$ be an $M$-bundle over a closed $3$-manifold $B$. A semi-$s$-cobordism from $B$ to $B'$ is a bordism $(V,B,B')$ with…

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…

Hopkins' extension conjecture. The map $\sigma(\mathbf{C})$ should be the restriction to $MU\langle 6\rangle$ of a similar map

Pull-back conjecture. The square

Let $n,m\geq0$ satisfy $d+1=2n+3m$ with $d\geq3$, and let $A_{d-1}\in Z^{d-1}(M_{d+1};\mathbb Z_2)$ and $B_2\in Z^2(M_{d+1};\mathbb Z_2)$ be cocycles. Consider the QCA family whose…

Let $S(X)=\frac{\sigma(X)}{8}\pmod{1}$ be the invertibly normalized response of the seven-dimensional generalized-semion QCA, where $\sigma(X)$ is the signature of a closed oriente…

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

An almost flat manifold is a closed manifold admitting metrics of arbitrarily small sectional curvature and uniformly bounded diameter; a flat manifold is a closed manifold with a…

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…

Spherical framed bordism comparison conjecture. There are natural isomorphisms of multiplicative homology theories compatible with the forgetful maps, making the diagram

Splitting conjecture. For every twice stabilised structure $\xi$ and every odd dimension $n$, the SKK sequence is split.

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…

Projection-splitting conjecture. It has been conjectured that this map is only one of many projection maps decomposing $MString$ into a direct sum of variants of $tmf$ and spectra…

Baas–Sullivan singularity conjecture. The resulting category is equivalent to the homotopy category of perfect $R$-modules. This would extend the proposed $MO$-module description o…

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…

Consider the bordism class represented by $S^8$ with the map to $B\mathrm{String}$ given by a generator of $[S^8,B\mathrm{String}]=\pi_8(B\mathrm{String})\cong\mathbb{Z}$, together…

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 $\mathbb Z_2$…

Consider Conner–Floyd's notion of bordism for manifolds with boundary, in which the Möbius strip is nonbounding. Conner–Floyd bordism conjecture. This is the correct kind of bordis…

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