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A compact manifold admits a Lorentzian structure iff its Euler characteristic vanishes. Therefore, every noncompact manifold admits a Lorentzian structure.

A surface which is homeomorphic to a finite collection of spheres, each with a finite number of handles, cross-handles, cross-caps, and perforations. A preliminary version of ...

In a space E equipped with a symmetric, differential k-form, or Hermitian form, the orthogonal sum is the direct sum of two subspaces V and W, which are mutually orthogonal. ...

A paracompact space is a T2-space such that every open cover has a locally finite open refinement. Paracompactness is a very common property that topological spaces satisfy. ...

A topological space X is pathwise-connected iff for every two points x,y in X, there is a continuous function f from [0,1] to X such that f(0)=x and f(1)=y. Roughly speaking, ...

The portion of a surface left when an open disk is removed from it.

The plumbing of a p-sphere and a q-sphere is defined as the disjoint union of S^p×D^q and D^p×S^q with their common D^p×D^q, identified via the identity homeomorphism. This ...

A nonsimply connected 3-manifold, also called a dodecahedral space.

A Möbius strip with finite thickness.

A surface constructed by placing a family of figure-eight curves into R^3 such that the first and last curves reduce to points. The surface has parametric equations x(u,v) = ...