The chess conjecture.
Virtual Legendrian isotopy
An elementary stabilization of a Legendrian knot L in the spherical cotangent bundle ST*M of a surface M is a surgery that results in attaching a handle to M along two discs away from the image in M of the projection of the knot L. A virtual Legendrian isotopy is a composition of stabilizations, destabilizations and Legendrian isotopies. A class of virtual Legendrian isotopy is called a virtual Legendrian knot. In contrast to Legendrian knots, virtual Legendrian knots enjoy the...
On the connectivity of finite subset spaces
We prove that the space of nonempty subsets of cardinality at most k in a bouquet of m+1-dimensional spheres is (m+k-2)-connected. This, as shown by Tuffley, implies that the space is (m+k-2)-connected for any m-connected cell complex X.
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