Obstruction Theory

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Obstruction is a concept in homotopy theory where an invariant equals zero if a corresponding problem is solvable and is non-zero otherwise. Let Y be a space, and assume for convenience that Y is n-simple for every n, that is, the action of π1(y) on πn(y) is trivial for every n. Under this hypothesis we can forget about base points for homotopy groups, and any map ƒ: S → Y determines an element of πn(Y).

Let B be a complex and A a subcomplex. Write Xn for A U Bn, where Bn denotes the n-skeleton of B. Let σ be an (n + I)-cell of B which is not in A. Let g = gσ be the attaching map σ. = Sn → Xn ⊂ B.

Given a map ƒ: Xn → Y, denote by c(ƒ) the cochain in Cn+1 (B, A; πn(Y)) given by c(ƒ): σ → [f º gσ]. Then it is clear that ƒ may be extended over Xngσ σ iff f º gσ is null-homotopic, that is, iff c(ƒ)(σ) = 0, and therefore that ƒ can be extended over Xn+1 = A ∪ Bn+1 if the cochain c(ƒ) is the zero cochain. It is a theorem of obstruction theory that c(ƒ) is a cocycle. It is called the obstruction cocycle or “the obstruction to extending ƒ over Bn+1

There are two immediate applications. First, any map of an n-dimensional complex K into an n-connected space X is null-homotopic.

Take (B, A) = (K x I, K x i) and define ƒ:A → X by the given map K → X on one piece and a constant map on the other piece; then ƒ can be extended over B because the obstructions lie in the trivial groups πi(X).

Second, as a particular case, a finite-dimensional complex K is contractible iff πi(K) is trivial for all i < dim K.

Suppose ƒ, g are two maps Xn → Y which agree on Xn-1. Then for each n-cell of B which is not in A, we get a map Sn → Y by taking ƒ and g on the two hemispheres. The resulting cochain of Cn(B, A; πn(Y)) is called the difference cochain of ƒ and g, denoted d(ƒ, g).

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