Poincaré lemmaIn mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed p-form on an open ball in Rn is exact for p with 1 ≤ p ≤ n. The lemma was introduced by Henri Poincaré in 1886. Especially in calculus, the Poincaré lemma also says that every closed 1-form on a simply connected open subset in is exact. In the language of cohomology, the Poincaré lemma says that the k-th de Rham cohomology group of a contractible open subset of a manifold M (e.
Relative homologyIn algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace. Given a subspace , one may form the short exact sequence where denotes the singular chains on the space X. The boundary map on descends to and therefore induces a boundary map on the quotient.
Verdier dualityIn mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of Alexander Grothendieck's theory of Poincaré duality in étale cohomology for schemes in algebraic geometry. It is thus (together with the said étale theory and for example Grothendieck's coherent duality) one instance of Grothendieck's six operations formalism.
Thom spaceIn mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space. One way to construct this space is as follows. Let be a rank n real vector bundle over the paracompact space B. Then for each point b in B, the fiber is an -dimensional real vector space. Choose an orthogonal structure on E, a smoothly varying inner product on the fibers; we can do this using partitions of unity.
Brown's representability theoremIn mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the Hotc of pointed connected CW complexes, to the Set, to be a representable functor. More specifically, we are given F: Hotcop → Set, and there are certain obviously necessary conditions for F to be of type Hom(—, C), with C a pointed connected CW-complex that can be deduced from alone. The statement of the substantive part of the theorem is that these necessary conditions are then sufficient.
Crystalline cohomologyIn mathematics, crystalline cohomology is a Weil cohomology theory for schemes X over a base field k. Its values Hn(X/W) are modules over the ring W of Witt vectors over k. It was introduced by and developed by . Crystalline cohomology is partly inspired by the p-adic proof in of part of the Weil conjectures and is closely related to the algebraic version of de Rham cohomology that was introduced by Grothendieck (1963).
Frank AdamsJohn Frank Adams (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory. He was born in Woolwich, a suburb in south-east London, and attended Bedford School. He began research as a student of Abram Besicovitch, but soon switched to algebraic topology. He received his PhD from the University of Cambridge in 1956. His thesis, written under the direction of Shaun Wylie, was titled On spectral sequences and self-obstruction invariants.
Modulo (mathematics)In mathematics, the term modulo ("with respect to a modulus of", the Latin ablative of modulus which itself means "a small measure") is often used to assert that two distinct mathematical objects can be regarded as equivalent—if their difference is accounted for by an additional factor. It was initially introduced into mathematics in the context of modular arithmetic by Carl Friedrich Gauss in 1801. Since then, the term has gained many meanings—some exact and some imprecise (such as equating "modulo" with "except for").
Highly structured ring spectrumIn mathematics, a highly structured ring spectrum or -ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an -ring is called an -ring. While originally motivated by questions of geometric topology and bundle theory, they are today most often used in stable homotopy theory. Highly structured ring spectra have better formal properties than multiplicative cohomology theories – a point utilized, for example, in the construction of topological modular forms, and which has allowed also new constructions of more classical objects such as Morava K-theory.
General positionIn algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings. For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident).
