nilpotence theorem
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In
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 classify ...
, the nilpotence theorem gives a condition for an element in the
homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
of a
ring spectrum In stable homotopy theory, a ring spectrum is a spectrum ''E'' together with a multiplication map :''μ'': ''E'' ∧ ''E'' → ''E'' and a unit map : ''η'': ''S'' → ''E'', where ''S'' is the sphere spectrum. These maps have to satisfy a ...
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 In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 199 ...
by and proved by .


Nishida's theorem

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


References

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Open online version.
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Further reading


Connection of ''X(n)'' spectra to formal group laws
Homotopy theory Theorems in algebraic topology {{Topology-stub