FINITE CATEGORIES

NOTES FOR THE REU: JULY 29 SCRIBE: NILES JOHNSON

For clarity, we state first the definition of homotopy equivalence:

Definition 0.1. Given two spaces, X and Y, and two maps f : X → Y and g : Y → X, we say that f and g are inverse homotopy equivalences if both composites are homotopic to the identity maps. That is,

g ∘ f ≃ id_{X} and f ∘ g ≃ id_{Y}

Previously we defined a series of functors

Spaces → Simplicial Sets → Simplicial Abelian Groups → Chain Complexes

taking

(X ⟶^{f} Y) ↦ (S_{∗}X ⟶^{f} S_{∗}Y) ↦ (A_{∗}X ⟶^{f} A_{∗}Y) ↦ (C_{∗}X ⟶^{f} C_{∗}Y)

and we noted that the product X × I gives

X × I ↦ S_{∗}(X × I) ↦ A_{∗}X ⊗ I ↦ C_{∗}X ⊗ I

A chain homotopy is defined precisely so that a homotopy of topological spaces produces, under these functors, a chain homotopy of complexes. Then the remainder of the result is given by a theorem.

Theorem 0.2. If f_{∗} : C_{∗} → D_{∗} and g : D_{∗} → C_{∗} are chain homotopic maps, then f_{∗} and g_{∗} induce inverse isomorphisms on the homology groups H_{∗}(C) and H_{∗}(D).

Example 0.3. Suppose that the unique map X → pt. and an inclusion pt → X are inverse homotopy equivalences, where pt is the one-point space (i.e. X is contractible). Then H_{n}(X) = H_{n}(pt) = 0 for n ≠ 0, and H_{0}(X) = H_{0}(pt) = ℤ.

Exercise 0.4. Show that D^{n} = {x ∈ ℝ^{n}||x| ≤ 1} is contractible. Hint: consider the map h : D^{n} × I → D^{n} given by h(x, t) = tx.

The following proposition is one of the foundational tools for computing homology.

Proposition 0.5. A short exact sequence of chain complexes

0 → C′_{∗} ⟶^{f_{∗}} C_{∗} ⟶^{g_{∗}} C′′_{∗} → 0

gives rise to a long exact sequence of homology groups

⋯ → H_{n}(C′) ⟶^{f_{∗}} H_{n}(C) ⟶^{g_{∗}} H_{n}(C′′) ⟶^{∂} H_{n−1}(C′) → · · · .

Proof. The only difficult part of this theorem is paying attention long enough to check all of the details. The major first step is the definition of ∂, which we give now.

Suppose x′′ ∈ Z′′_{n}. By the surjectivity of g_{n}, ∃ x ∈ C_{n} such that g_{n}(x) = x′′. Now since g_{∗} commutes with the differentials, g_{n−1}d_{n}(x) = d′′_{n}g_{n}(x) = d′′_{n}(x′′) = 0. Therefore, by exactness at C_{∗}, ∃ x′ ∈ C′_{n−1} such that f_{n−1}(x′) = d_{n}(x). Now since f_{∗} commutes with the differentials, f_{n−1}d′_{n−1}(x′) = d_{n}f_{n−1}(x′) = d_{n}^{2}(x) = 0. So d′_{n−1}(x′) ∈ Z′_{n−1}. We define ∂(x′′) to be the homology class of d′_{n−1}(x′).

What remains now is to check that ∂ is a well-defined homomorphism of apelian groups, and that the sequence above is exact at all stages. None of these are particularly difficult, they do require patience and care and are (as always) left to the reader.

As an example of the usefulness of a long exact sequence, we note that one can immediately compute the homology of all spheres inductively. For this we introduce the reduced homology groups. For any space X, the reduced homology H̃_{n}(X) is defined so that H_{n}(X) = H̃_{n}(X) ⊕ ℤ, where ℤ is in degree zero. Then there is a similar long exact sequence for quotients; if A ⊂ X is a sub-simplicial complex, then we have a long exact sequence

⋯ → H̃_{n}(A) → H̃_{n}(X) → H̃_{n}(X/A) → H̃_{n−1}(A) → ⋯

Example 0.6. The n-sphere S^{n} is a quotient D^{n}/S^{n−1}, and applying the sequence above we have

⋯ → H̃_{q}(S^{n−1}) → H̃_{q}(D^{n}) → H̃_{q}(S^{n}) → H̃_{q−1}(S^{n−1}) → H̃_{q} − 1(D^{n}) ⋯

now recalling that H̃_{q}(D^{n}) = 0 ∀ n, we have that

H̃_{q}(S^{n}) ≅ H̃_{q−1}(S^{n−1}) for all q

Thus, inductively, H̃_{n}(S^{n}) ≅ ℤ and H̃_{q}(S^{n}) = 0 for q ≠ 0.

Remark 0.7. This same method can be applied to any space X and its suspension, ΣX, to find

H̃_{q}(ΣX) ≅ H̃_{q−1}(X) for all q.