The Taniyama-Shimura conjecture, since its proof now sometimes known as the modularity theorem, is very general and important conjecture (and now theorem) connecting topology and number theory which arose from several problems...
The universal cover of a connected topological space is a simply connected space with a map that is a covering map. If is simply connected, i.e., has a trivial fundamental group, then it is its own universal cover. For...
The homotopy groups generalize the fundamental group to maps from higher dimensional spheres, instead of from the circle. The th homotopy group of a topological space is the set of homotopy classes of maps from the n-sphere to ,...
Two mathematical objects are said to be homotopic if one can be continuously deformed into the other. For example, the real line is homotopic to a single point, as is any tree. However, the circle is not contractible, but is...
The fundamental group of an arcwise-connected set is the group formed by the sets of equivalence classes of the set of all loops, i.e., paths with initial and final points at a given basepoint , under the equivalence relation of...
A diagram lemma which states that, given the above commutative diagram with exact rows, the following holds: 1. If is surjective, and and are injective, then is injective; 2. If is injective, and and are...
A diagram lemma which states that, given the commutative diagram of additive Abelian groups with exact rows, the following holds: 1. If is surjective, and and are injective, then is injective; 2. If is...
he Euler numbers, also called the secant numbers or zig numbers, are defined for by (1) (2) where is the hyperbolic secant and sec is the secant. Euler numbers give the number of odd alternating...
