Topology

15 problems

A four-dimensional space can have the topology of a sphere. Must every smooth version be smoothly equivalent to the standard sphere?

For certain closed manifolds, does matching their homotopy structure force them to be the same topological shape?

Do a manifold’s higher signatures stay unchanged when its homotopy structure is preserved?

Does a complex hypersurface singularity’s local topological shape determine its multiplicity?

Can the proper-action geometry of a countable discrete group recover the K-theory of its reduced operator algebra?

Does every separable nuclear C*-algebra satisfy the universal coefficient theorem relating its K-theoretic invariants?

Does every contractible two-dimensional polyhedron become collapsible after taking its product with one interval?

Can algebraic invariants of every group be assembled from its virtually cyclic subgroups?

Can finite groups detect every nontrivial element of every hyperbolic group?

Does a hyperbolic group whose boundary is a two-sphere act geometrically on hyperbolic three-space?

Can the topology of a finite group’s elementary abelian 2-subgroups detect its normal 2-subgroup structure?

Does nonnegative sectional curvature force finite total rational homotopy in a closed simply connected manifold?

Can two distinct surgery slopes on the same nontrivial knot produce the same oriented three-manifold?

Does exponential growth of colored Jones evaluations recover the volume of every hyperbolic knot complement?

Can surgery on a knot create a genuinely decomposable 3-manifold only through the known cable-knot construction?