Poincaré space
Encyclopedia
In algebraic topology
, a Poincaré space is an n-dimensional topological space
with a distinguished element µ of its nth homology group such that taking the cap product
with an element of the kth cohomology
group yields an isomorphism to the (n − k)th cohomology group. The space is essentially one for which Poincaré duality
is valid.
For example, any closed, orientable, connected manifold is a Poincaré space.
Sometimes, Poincaré space means a homology sphere
—for instance, the Poincaré dodecahedral space in 3 dimensions.
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...
, a Poincaré space is an n-dimensional topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
with a distinguished element µ of its nth homology group such that taking the cap product
Cap product
In algebraic topology the cap product is a method of adjoining a chain of degree p with a cochain of degree q, such that q ≤ p, to form a composite chain of degree p − q. It was introduced by Eduard Čech in 1936, and independently by Hassler Whitney in 1938.-Definition:Let X be a topological...
with an element of the kth cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
group yields an isomorphism to the (n − k)th cohomology group. The space is essentially one for which Poincaré duality
Poincaré duality
In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
is valid.
For example, any closed, orientable, connected manifold is a Poincaré space.
Sometimes, Poincaré space means a homology sphere
Homology sphere
In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1. That is,andTherefore X is a connected space, with one non-zero higher Betti number: bn...
—for instance, the Poincaré dodecahedral space in 3 dimensions.