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A branch of mathematics which brings together ideas from algebraic geometry, linear algebra, and number theory. In general, there are two main types of K-theory: topological ...

A map u:M->N, between two compact Riemannian manifolds, is a harmonic map if it is a critical point for the energy functional int_M|du|^2dmu_M. The norm of the differential ...

Chain equivalences give an equivalence relation on the space of chain homomorphisms. Two chain complexes are chain equivalent if there are chain maps phi:C_*->D_* 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, ...

An iterated fibration of Eilenberg-Mac lane spaces. Every topological space has this homotopy type.

The Stiefel manifold of orthonormal k-frames in R^n is the collection of vectors (v_1, ..., v_k) where v_i is in R^n for all i, and the k-tuple (v_1, ..., v_k) is ...

An H-space, named after Heinz Hopf, and sometimes also called a Hopf space, is a topological space together with a continuous binary operation mu:X×X->X, such that there ...

The bipartite double graph, also called the Kronecker cover, Kronecker double cover, bipartite double cover, canonical double cover, or bipartite double, of a given graph G ...

The fundamental group of an arcwise-connected set X is the group formed by the sets of equivalence classes of the set of all loops, i.e., paths with initial and final points ...

Milnor (1956) found more than one smooth structure on the seven-dimensional hypersphere. Generalizations have subsequently been found in other dimensions. Using surgery ...