Math Problems Vault

Problem 6.13.1

With $\epsilon$ , $H_0(C)$ , $\bar{H}_0(C)$ as defined below, show that the short exact sequence

0 \to \ker (\epsilon ) \to C_0 \xrightarrow {\epsilon } \mathbb {Z} \to 0,

leads to a short exact sequence

0 \to \bar{H}_0(C) \to H_0(C) \to \mathbb {Z} \to 0,

which splits to give $H_0(C) \simeq \bar{H}_0(C) \oplus \mathbb {Z}$ .

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Remark.

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Given a chain complex of free abelian groups $(C_i, \partial_i)$ , $i \geq 0$ , extend it by introducing $C_{-1} = \mathbb {Z}$ . Define $\partial_0 = \epsilon$ by first selecting a basis of generators of $C_0$ and setting $\epsilon (g) = 1$ for each generator $g$ ; then $\ker (\epsilon )$ is generated as a free group on differences of generators. When $\partial_1$ has the property that, for a generator $g$ of $C_1$ , $\partial_1(g)$ is a difference of generators, we have $\mathrm{im}(\partial_1) \subset \ker (\epsilon )$ , so the extended chain complex is still a chain complex; $\partial_0 = \epsilon$ is called the augmentation.

Definition 6.13.1: the reduced homology $\bar{H}_k(C)$ is the homology of the augmented chain complex for $k \geq 0$ . This agrees with $H_k(C)$ in dimensions greater than $0$ , but $\bar{H}_0(C) = \ker (\epsilon )/\mathrm{im}(\partial_1)$ , whereas $H_0(C) = C_0/\mathrm{im}(\partial_1)$ .

For the singular complex $S(X)$ of a topological space $X$ , the augmentation $\epsilon : S_0(X) \to \mathbb {Z}$ is defined by $\epsilon (\sigma ) = 1$ for a singular $0$ -simplex $\sigma$ , and the reduced homology $\bar{H}_k(X)$ is the homology of this augmented singular complex.

Problem 6.13.2

(a)(b)(c)

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(a)

Verify that for $S(X)$ we have $\epsilon \partial_1 = 0$ (notation as in Exercise 6.13.1).

(b)

Show that $H_0(X) \simeq \bar{H}_0(X) \oplus \mathbb {Z}$ .

(c)

Show that a topological space is path connected iff $\bar{H}_0(X) \simeq 0$ .

Problem 6.13.3

Show that the reduced homology of the sphere can be expressed as

\bar{H}_k(S^n) = \begin{cases} \mathbb {Z} & k = n, \\ 0 & k \neq n. \end{cases}

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Problem 6.13.4

Show that the homology of the suspension $\Sigma X$ of a path-connected space can be expressed as $H_{k+1}(\Sigma X) \simeq \bar{H}_{k+1}(\Sigma X) = \bar{H}_k(X)$ for $k \geq 0$ .

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Problem 6.13.5

By using the augmented complex in place of the original complex and using $S_{-1}(X, A) = S_{-1}(X)/S_{-1}(A) = \mathbb {Z}/\mathbb {Z} = 0$ , show that there is a long exact sequence in reduced homology which ends in

H_1(X, A) \to \bar{H}_0(A) \to \bar{H}_0(X) \to H_0(X, A) \to 0.

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Problem 6.13.6

Assume that $X = A \cup B$ , $A \cap B \neq \emptyset$ , and the hypotheses giving the usual Mayer--Vietoris exact sequence hold. The Mayer--Vietoris exact sequence can be modified to use reduced homology as long as $A \cap B \neq \emptyset$ , augmenting the usual singular chains by $S_{-1}(A \cap B) = \mathbb {Z}$ , $S_{-1}(A) \oplus S_{-1}(B) = \mathbb {Z} \oplus \mathbb {Z}$ , $S_{-1}(A) + S_{-1}(B) = \mathbb {Z}$ , $S_{-1}(X) = \mathbb {Z}$ .

Show that there is a long exact sequence using reduced homology which ends in

H_1(X) \xrightarrow {\delta } \bar{H}_0(A \cap B) \xrightarrow {i_0} \bar{H}_0(A) \oplus \bar{H}_0(B) \xrightarrow {j_0} \bar{H}_0(X) \to 0.

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Problem 6.13.7

Use the reduced Mayer--Vietoris sequence of Exercise 6.13.6 to show that, if $A$ and $B$ are path connected, $A \cap B \neq \emptyset$ , $A \cup B = X$ , and the hypotheses giving the Mayer--Vietoris exact sequence apply, then $X$ is path connected.

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