HOMOTOPY THEORY

ADAM KAYE

1. CW Approximation

The CW approximation theorem says that every space is weakly equivalent to a CW complex.

Theorem 1.1 (CW Approximation). There exists a functor Γ from the homotopy category of spaces to itself such that for every space X, ΓX is a CW complex and there is a natural transformation γ : Γ → id such that γ : ΓX → X is a weak equivalence. If X is n-connected then ΓX has one vertex and no q-cells for 1 ≤ q ≤ n.

Proof. Without loss of generality we may assume X is connected. The inclusion of a point in X then gives a surjection on π_{0}(X). We inductively define X_{q} and γ_{q} : X_{q} → X. Assume we have a CW complex X_{q} and γ_{q} that induces an surjection on π_{q} and an isomorphism on the lower homotopy groups. We construct an new space X′_{q+1} by attaching a q + 1 dimensional disk D_{f}^{q+1} along a map f for each f ∈ π_{q}(X) in the kernel of γ_{q}. Since γ_{q∗}(f) is nullhomotopic we can extend γ_{q} to X′_{q+1} by defining it on D_{f}^{q+1} as the nullhomotopy of f. Now π_{q}(X′_{q+1}) is isomorphic to π_{q}(X).

We then construct X_{q+1} as

X_{q+1} = X′_{q+1} ∨ ⋁_{j∈A} S_{j}^{q+1}

where A is a generating subset of π_{q+1}(X). We then define γ_{q+1} : X_{q+1} → X by defining it as j on S_{j}^{q+1} and the extension of γ_{q} on X′_{q+1}. Thus π_{q+1}(X_{q+1}) surjects onto π_{q+1}(X). Cellular approximation of maps shows that this construction does not change π_{i} for i < q because π_{i} only depends on the i + 1 skeleton which hasn’t been altered. Thus we now have a sequence of inclusions X_{0} ↪ X_{1} ↪ X_{2} ↪ … and maps γ_{q} : X_{q} → X such that for each q, γ_{q} induces an isomorphism on π_{q−1} and a surjection on π_{q}. Let ΓX be the colimit of the X_{q} and γ be the map ΓX → X induced by the γ_{q}. From the construction we can see that if X is n-connected then X_{q} is just a vertex for each q ≤ n and so ΓX has no q-cells for q ≤ n.

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2. Homotopy Excision

Definition 2.1. Given spaces A, B ⊂ X, (X; A, B) is called an excisive triad if X is contained in the union of the interiors of A and B.

Date: 7/30/10.

The excision theorem for homology says that the inclusion of pairs (A, A∩B) ↪ (X, B) induces an isomorphism on homoology groups. We would like to have a similar theorem for homotopy groups, but unfortunately we only have an isomorhpism for the lower homotopy groups.

Definition 2.2. f : X → Y is an n-equivalence if it induces an isomorphism on π_{q} for q < n and an surjection on π_{n}. Thus a map is a weak equivalence if and only if it is an n-equivalence for all n.

Theorem 2.1 (Homotopy Excision). If (X; A, B) is an excisive triad with (A, A ∩ B) m-connected and (B, A ∩ B) n-connected then (A, A ∩ B) ↪ (X, B) is a m + nequivalence.

The proof can be found in Allen Hatcher’s Algebraic Topology (page 361) or J. P. May’s A Concise Course in Algebraic Topology (page 86).

Corollary 2.2. If f : X → Y is an n-equivalence with X, Y (n − 1)-connected then p : (Mf, X) → (Cf, ∗) is a 2n-equivalence. Here Mf is the mapping cylinder (X × [0, 1] ∪ Y)/{(x, 1) ∼ f(x)} and X is considered a subspace of Mf under the inclusion x ↦ (x, 0). Cf is the mapping cone which is given by identifying the image of X in Mf to point and p is the natural map that sends (x, 0) to the basepoint and is the identity on everything else.

Proof. Consider the subsets of Cf defined by A = Y ∪ {(x, t) : 0 ≤ t ≤ 2/3}, B = {(x, t) : 1/3 ≤ t ≤ 1}. Then (Cf; A, B) form an excisive triad and A ∩ B ≃ X. We can see that (A, A ∩ B) is n-connected because f_{∗} maps π_{n}(X) onto π_{n}(Y) and therefore every element of π_{n}(A) is represented by an element of π_{n}(X). Thus every element of π_{n}(A) is homotopic to an element of π_{n}(A ∩ B) which implies π_{n}(A, A ∩ B) = 0 and since X, Y are (n − 1)-connected and we have just shown π_{n}(A, A ∩ B) = 0 we may conclude that (A, A ∩ B) is n-connected. Letting CX = X × [0, 1]/{(x, 1) ≃ ∗} denote the cone on X with the natural inclusion X ↪ CX x ↦ (x, 0) we get the long exact sequence

⋯ → π_{q}(CX) → π_{q}(CX, X) → π_{q−1}(X) → π_{q−1}(CX) → ….

Since CX is contractible it has trivial homotopy groups which implies π_{q}(CX, X) ≅ π_{q−1}(X) and the fact that X is (n − 1)-connected implies (CX, X) is n-connected. Since (B, A ∩ B) is homotopic to (CX, X), (X, A ∩ B) is n-connected. Thus the homotopy excision theorem implies that the natural map (A, A ∩ B) → (Cf, B) is a 2n-equivalence. We also have homotopy equivalences (Mf, X) ≃ (A, A ∩ B) and (Cf, B) ≃ (Cf, ∗) and p is given by the composite (Mf, X) → (A, A ∩ B) → (Cf, B) → (Cf, ∗). Since each map is a 2n-equivalence or a homotopy equivalence the composition is a 2n-equivalence. □

Corollary 2.3. Let i : A → X be a cofibration and an n-equivalence between (n − 1)-connected spaces. Then the natural map q : (X, A) → (X/A, ∗) is a 2nequivalence.

Proof. The previous corollary implies that p : (Mi, A) → (Ci, ∗) is a 2n-equivalence. (Mi, A) is homotopy equivalent to (X, A) and (Ci, ∗) is homotopy equivalence to (X/A, ∗) because the inclusion of A is a cofibration. q is given by the composition

(X, A) →^{≃} (Mi, A) →^{p} (Ci, ∗) →^{≃} (X/A, ∗)

so it is a 2n-equivalence. □

3. Freudenthal Suspension

For each q there is a map Σ : π_{q}(X) → π_{q+1}(ΣX) defined by f ↦ f ∧ id : S^{q} ∧ S^{1} → X ∧ S^{1}. This is an element of π_{q+1}(ΣX) because S^{q} ∧ S^{1} ≅ S^{q+1}.

Theorem 3.1 (Freudenthal Suspension). If X is (n − 1)-connected and the inclusion of the base point ∗ ↪ X is a cofibration then Σ induces an isomorphism on π_{q} for q < 2n − 1 and a surjection for q = 2n − 1.

Proof. Write ΣX as C_{−}X ∪ C_{+}X where the − and + denote two different copies of X ∧ [0, 1] glued along {(x, 0)} (here we are using the reduced cone, as opposed to the unreduced cone used earlier). Then C_{−}X ∩ C_{+}X = X. The same argument as before shows that π_{i}(X) ≅ π_{i+1}(C_{+}X, X). Since C_{−}X is contractible, π_{i}(ΣX, C_{−}X) ≅ π_{i}(ΣX). Furthermore, Σ is given by the composition

π_{i}(X) → π_{i+1}(C_{+}X, X) → π_{i+1}(ΣX, C_{−}X) → π_{i+1}(ΣX)

because the first map is f ↦ f ∧ [0, 1]. We would like to apply homotopy excision to the second map. Note that to do this we actually need to replace C_{+}X, C_{−}X with open neighborhoods of C_{+}X, C_{−}X so that (ΣX; C_{−}X, C_{+}X) is an excisive triad. Since X is (n − 1)-connected, (C_{±}X, X) is n-connected (again because π_{i+1}(CX, X) ≅ π_{i}(X)) and this implies that the second map is an isomorphism for i + 1 < 2n and a surjection for i + 1 = 2n. Thus Σ induces an isomorphism π_{q}(X) → π_{q+1}(ΣX) for q < 2n − 1 and a surjection for i = 2n − 1. □

The Freudenthal suspension Theorem has one very important application in that it allows us to calculate some homotopy groups, namely π_{n}(S^{n}). We have seen that the Hopf fibration S^{1} → S^{3} → S^{2} gives a long exact sequence

⋯ → π_{2}(S^{3}) → π_{2}(S^{2}) → π_{1}(S^{1}) → π_{1}(S^{3}) → ….

Since π_{i}(S^{3}) = 0 for i = 1, 2 we have π_{1}(S^{1}) ≅ π_{2}(S^{2}). We have also already calculated π_{1}(S^{1}) ≅ ℤ. Freudenthal suspension implies that Σ : π_{q}(S^{q}) → π_{q+1}(S^{q+1}) is an isomorphism for q ≥ 2 so we may conclude that π_{n}(S^{n}) ≅ ℤ for all n.

It also helps to calculate stable homotopy groups. These are defined as π_{q}^{s}(X) = colimπ_{q+n}(Σ^{n}X). Freudenthal suspension implies that the sequence π_{q}(X) → π_{q+1}(ΣX) → π_{q+2}(Σ^{2}X) → … is eventually constant and this is in fact where the term stable homotopy group comes from.

4. Hurewicz

There is an isomorphism H̃_{i}(X) → H̃_{i+1}(ΣX) which we also denote by Σ. The long exact sequence

⋯ → H̃_{i+1}(CX) → H_{i+1}(CX, X) → H̃_{i}(X) → H̃_{i}(CX) → …

gives an isomorphism H̃_{i} → H_{i+1}(CX, X) because H̃_{i}(CX) = 0 for all i. Since H_{i}(CX, X) ≅ H̃_{i}(ΣX) we can set Σ equal to the composition

H̃_{i}(X) → H_{i+1}(CX, X) → H̃_{i+1}(X).

We fix generators i_{n} for H̃_{n}(S^{n}) such that Σi_{n} = i_{n+1}. We then construct h : π_{n}(X) → H̃_{n}(X) by h([f]) = f_{∗}(i_{n}) (since f is a map S^{n} → X it induces a map f_{∗} : H̃_{n}(S^{n}) → H̃_{n}(X)). This is a homomorphism because [f + g] is given by the composition

S^{n} → S^{n} ∨ S^{n} →^{f∨g} X ∨ X → X

where the first map is the map that sends one hemisphere to the first copy of S^{n} and the other hemisphere to the other copy of S^{n}. Then the induced map on H̃_{n} sends i_{n} ↦ i_{n} ⊕ i_{n} ↦ f_{∗}(i_{n}) ⊕ g_{∗}(i_{n}) ↦ f_{∗}(i_{n}) + g_{∗}(i_{n}) so h is in fact a homomorphism. In the case where n = 1 the fact that H̃_{n}(X) is abelian implies that h induces a map from π′_{1}(X) (the abelianization of π_{1}(X)) to H̃_{1}(X). We can now state the Hurewicz theorem which gives a connection between homology groups and homotopy groups.

Theorem 4.1 (Hurewicz). If X is (n − 1)-connected then h is an isomorphism if n > 1 and if n = 1 the induced map on π′_{1}(X) → H̃_{1}(X) is an isomorphism.

Proof. If X is a sphere then h([id]) = i_{n} and our previous calculation for π_{n}(S^{n}) shows that the theorem holds in this case. In the general case we may assume that X is a CW-complex with 1 vertex and no q-cells for q < n by the CW approximation theorem. Cellular approximation of maps implies that the inclusion X^{n+1} ↪ X is an n+1-equivalence. It is easy to see by the CW construction of homology that this inclusion induces an isomorphism on H̃_{n} as well because H̃_{n} is determined solely by the terms of the chain complex in degrees n and n + 1. Thus we may assume without loss of generality that X = X^{n+1}.

Let L = X^{n} and let K = ∨_{α∈A}S_{α}^{n} where A is the set of n + 1 cells in X. Define f : K → L by letting it equal the attaching map of α on S_{α}^{n}. Then X is the mapping cone of f (unreduced). The maps K →^{f} L ↪ X induces maps on homology and homotopy and our goal is to show that the rows in the commutative diagram

\xymatrix{π_{n}(K) \ar[r] \ar[d] & π_{n}(L) \ar[r] \ar[d] & π_{n}(X) \ar[r] \ar[d] & 0 \\ H̃_{n}(K) \ar[r] & H̃_{n}(L) \ar[r] & H̃_{n}(X) \ar[r] & 0}

are exact. Since K, L are wedges of spheres the theorem holds for them and we can then apply the five lemma to get the result for X. The diagram is commutative because h is natural. This comes from the fact that if g : X → Y is a map of spaces and σ : S^{n} → X is an element of π_{n}(X) then (f ∘ σ)_{∗}(i_{n}) = f_{∗}(σ_{∗}(i_{n})). H̃_{n}(L) is the free abelian group generated by the n-cells and a linear compbination of n-cells maps to 0 in H̃_{n}(X) if and only if it is the boundary of discs. Since the boundaries of discs are given precisely by the image of K we have exactness of the H̃ sequence at L. The cokernel of the map H̃_{n}(L) → H̃_{n}(X) should be H_{n}(X, L) = H̃_{n}(X/L), but X/L is a wedge of n + 1 spheres so it has trivial H̃_{n}. Thus the bottom row is exact.

For n > 1 we may write f as the composition K → Mf →^{p} L where p is the natural homotopy equivalence. This gives us the commutative diagram

\xymatrix{π_{n}(K) \ar[r] \ar[d]^{id} & π_{n}(Mf) \ar[r] \ar[d]^{p_{∗}} & π_{n}(Mf, K) \ar[r] \ar[d] & 0 \\ π_{n}(K) \ar[r] & π_{n}(L) \ar[r] & π_{n}(X) \ar[r] & 0}

in which the two left vertical arrows are isomoprphisms. The top row is exact because the cokernel of the last map is π_{n−1}(K) and this is trivial because K is a wedge of n spheres. The right vertical arrow is an isomorphism by the first corollary to the homotopy excision theorem. Thus the bottom row is exact and we are done with the case n > 1. In the case with n = 1 the top row is not exact because the kernel of the map π_{1}(Mf) → π_{1}(Mf, K) is the normal subgroup generated by π_{1}(K). But then it becomes exact when we take the abelianization of the whole diagram and this gives us the n = 1 case.

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References

[1] May, J. P. A Concise Course in Algebraic Topology. University of Chicago Press. 1999.

[2] Hatcher, Allen. Algebraic Topology. Cambridge University Press. 2002.