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Let X be a continuum (i.e., a compact connected metric space). Then X is hereditarily unicoherent provided that every subcontinuum of X is unicoherent. Any hereditarily ...

Let C denote a chain complex, a portion of which is shown below: ...->C_(n+1)->C_n->C_(n-1)->.... Let H_n(C)=kerpartial_n/Impartial_(n+1) denotes the nth homology group. Then ...

One of the Eilenberg-Steenrod axioms which states that, if f:(X,A)->(Y,B) is homotopic to g:(X,A)->(Y,B), then their induced maps f_*:H_n(X,A)->H_n(Y,B) and ...

The Hopf invariant one theorem, sometimes also called Adams' theorem, is a deep theorem in homotopy theory which states that the only n-spheres which are H-spaces are S^0, ...

A homotopy from one embedding of a manifold M in N to another such that at every time, it is an embedding. The notion of isotopy is category independent, so notions of ...

Given a map f from a space X to a space Y and another map g from a space Z to a space Y, does there exist a map h from X to Z such that gh=f? If such a map h exists, then h ...

If a sphere is covered by three closed sets, then one of them must contain a pair of antipodal points.

In abstract topology, a machine is method for producing infinite loop spaces and spectra. In automata theory, an abstract machine that is implemented in hardware is simply ...

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

If a compact manifold M has nonnegative Ricci curvature tensor, then its fundamental group has at most polynomial growth. On the other hand, if M has negative curvature, then ...