Formal manifold
Encyclopedia
In geometry
and topology
, a formal manifold can mean one of a number of related concepts:
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a formal manifold can mean one of a number of related concepts:
- In the sense of Dennis SullivanDennis SullivanDennis Parnell Sullivan is an American mathematician. He is known for work in topology, both algebraic and geometric, and on dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University.-Work in topology:He...
, a formal manifold is one whose real homotopy type is a formal consequence of its real cohomology ringCohomology ringIn mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is...
; algebro-topologically this means that all Massey productMassey productIn algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product.-Massey triple product:...
s vanish. - A stronger notion is a geometrically formal manifold, which is the condition that all wedge products of harmonic forms are harmonic.