Nilpotence theorem
Encyclopedia
In algebraic topology
, the nilpotence theorem gives a condition for an element of the coefficient ring
of a ring spectrum to be nilpotent
, in terms of complex cobordism
. It was conjectured by and proved by .
are nilpotent. This is a special case of the nilpotence theorem.
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, the nilpotence theorem gives a condition for an element of the coefficient ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
of a ring spectrum to be nilpotent
Nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
, in terms of complex cobordism
Complex cobordism
In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories...
. It was conjectured by and proved by .
Nishida's theorem
showed that elements of positive degree of the homotopy groups of spheresHomotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting...
are nilpotent. This is a special case of the nilpotence theorem.