Localization of a topological space
Encyclopedia
In mathematics, well behaved topological spaces can be localized at primes, in a similar way to the localization of a ring
at a prime. This construction was described by Dennis Sullivan
in 1970 lecture notes that were finally published in .
The reason to do this was in line with an idea of making topology, more precisely algebraic topology
, more geometric. Localization of a space X is a geometric form of the algebraic device of choosing 'coefficients' in order to simplify the algebra, in a given problem. Instead of that, the localization can be applied to the space X, directly, giving a second space Y.
. Then there is a simply connected CW complex Y together with a map from X to Y such that
This space Y is unique up to homotopy equivalence, and is called the localization
of X at A.
If A is the localization of Z at a prime p, then the space Y is called the localization of X at p
The map from X to Y induces isomorphisms from the A-localizations of the homology
and homotopy groups of X to the homology
and homotopy groups of Y.
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
at a prime. This construction was described by Dennis Sullivan
Dennis Sullivan
Dennis 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...
in 1970 lecture notes that were finally published in .
The reason to do this was in line with an idea of making topology, more precisely algebraic topology
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...
, more geometric. Localization of a space X is a geometric form of the algebraic device of choosing 'coefficients' in order to simplify the algebra, in a given problem. Instead of that, the localization can be applied to the space X, directly, giving a second space Y.
Definitions
We let A be a subring of the rational numbers, and let X be a simply connected CW complexCW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
. Then there is a simply connected CW complex Y together with a map from X to Y such that
- Y is A-local; this means that all its homology groups are modules over A
- The map from X to Y is universal for (homotopy classes of) maps from X to A-local CW complexes.
This space Y is unique up to homotopy equivalence, and is called the localization
of X at A.
If A is the localization of Z at a prime p, then the space Y is called the localization of X at p
The map from X to Y induces isomorphisms from the A-localizations of the homology
and homotopy groups of X to the homology
and homotopy groups of Y.