Hatcher Exercise 2.2.28

(a) Let be the space obtained from the torus with a Mobius band attached by gluing the its single boundary component to . We compute the homology using Meyer-Vietoris. As usual, let be with a small neighborhood inside , and the Mobius band with a small neighborhood inside .

0 \rightarrow \wt{H}_2(A) \oplus \wt{H}_2(B) \rightarrow \wt{H}_2(X) \rightarrow \wt{H}_1(A \cap B) \rightarrow \wt{H}_1(A) \oplus \wt{H}_1(B) \rightarrow \wt{H}_1(X) \rightarrow 0 \dots

0 \rightarrow \mathbb{Z} \rightarrow \wt{H}_2(X) \xrightarrow{\delta} \la a \ra \xrightarrow{\Phi} \la a,b,c \ra \xrightarrow{\Psi} \wt{H}_1(X) \xrightarrow{\delta} 0

\wt{H}_1(X) \cong \dfrac{\la a,b,c \ra}{\im \Phi} = \dfrac{\la a,b,c \ra}{\la a-2c \ra} = \dfrac{\la a-2c,b-2c,c \ra}{\la a-2c \ra} \cong \mathbb{Z}^2

0 \rightarrow \wt{H}_2(X) \rightarrow \la a \ra \xrightarrow{\Phi} \dfrac{\la a,b \ra}{\la 2a \ra} \xrightarrow{\Psi} \wt{H}_1(X) \rightarrow 0

\wt{H}_1(X) \cong \dfrac{\la a,b \ra}{\la 2a, a-2b \ra} = \dfrac{\la a+2b, b \ra}{\la a+2b, -4b \ra} \cong \mathbb{Z}_4

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