approximation theorem for an arbitrary space
Theorem 0.1.
(Approximation theorem for an arbitrary topological space in terms of the colimit
of a sequence
of cellular inclusions of CW-complexes):
“There is a functor
Γ:𝒉𝑼⟶𝒉𝑼 where hU is the homotopy category for unbased spaces , and a natural transformation γ:Γ⟶Id that asssigns a CW-complex ΓX and a weak equivalence
γe:ΓX⟶X to an arbitrary space X, such that the following diagram commutes:
ΓXΓf→ΓY γ(X)↓↓γ(Y)X@>f≫Y and Γf:ΓX→ΓY is unique up to homotopy equivalence
.”
(viz. p. 75 in ref. [1]).
Remark 0.1.
The CW-complex specified in the approximation theorem for an arbitrary space (http://planetmath.org/ApproximationTheoremForAnArbitrarySpace) is constructed as the colimit ΓX of a sequence of cellular inclusions of CW-complexes X1,…,Xn , so that one obtains X≡colim[Xi]. As a consequence of J.H.C. Whitehead’s Theorem, one also has that:
γ*:[ΓX,ΓY]⟶[ΓX,Y] is an isomorphism.
Furthermore, the homotopy groups of the CW-complex ΓX are the colimits of the
homotopy groups of Xn and γn+1:πq(Xn+1)⟼πq(X) is a group epimorphism.
References
- 1 May, J.P. 1999, A Concise Course in Algebraic Topology., The University of Chicago Press: Chicago
Title | approximation theorem for an arbitrary space |
Canonical name | ApproximationTheoremForAnArbitrarySpace |
Date of creation | 2013-03-22 18:14:40 |
Last modified on | 2013-03-22 18:14:40 |
Owner | bci1 (20947) |
Last modified by | bci1 (20947) |
Numerical id | 43 |
Author | bci1 (20947) |
Entry type | Theorem |
Classification | msc 81T25 |
Classification | msc 81T05 |
Classification | msc 81T10 |
Classification | msc 55U15 |
Classification | msc 57Q05 |
Classification | msc 57Q55 |
Classification | msc 55U05 |
Classification | msc 55U10 |
Synonym | approximation theorems for topological spaces |
Related topic | TheoremOnCWComplexApproximationOfQuantumStateSpacesInQAT |
Related topic | CWComplex |
Related topic | SpinNetworksAndSpinFoams |
Related topic | HomotopyCategory |
Related topic | WeakHomotopyEquivalence |
Related topic | GroupHomomorphism |
Related topic | ApproximationTheoremAppliedToWhitneyCrMNSpaces |
Defines | unique colimit of a sequence of cellular inclusions of CW-complexes |