algebraic topology Algebraic topology is a branch of mathematics that 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 classif ...

, the nilpotence theorem gives a condition for an element in the homotopy groups of a ring spectrum to be

nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...

, in terms of the complex cobordism spectrum

\mathrm

. More precisely, it states that for any ring spectrum

R

, the kernel of the map

\pi_\ast R \to \mathrm_\ast(R)

consists of nilpotent elements. It was conjectured by and proved by .

Nishida's theorem

showed that elements of positive degree of the homotopy groups of spheres are nilpotent. This is a special case of the nilpotence theorem.

References

* * .
Open online version.
*

Nishida's theorem

References

Further reading