homology group of connected sum of Klein bottle and $RP^2$ How to compute the homology group of connected sum of Klein bottle and $RP^2$?
I want to use mayer vietories sequence,I know Klein bottle remove a disk is $S^1 \vee S^1$，and $RP^2$ remove a disk is $S^1 $,but i can't write the generator.
 A: So you have constructed the following long exact sequence but are unsure what the map $f$ is.
$$
\begin{array}{cccccc}
&0  &\to& 0 &\to& H_2(K\#\mathbb{R}P^2)\\
\to& H_1(S^1)\cong\mathbb{Z}  &\stackrel f\to& H_1(S^1\vee S^1)\oplus H_1(S^1)\cong \mathbb{Z}^3 &\to& H_1(K\#\mathbb{R}P^2)\\\stackrel 0\to& \mathbb{Z}&\to& \mathbb{Z}\oplus \mathbb{Z}&\to& \mathbb{Z}\\\to&0
\end{array}
$$
As the inclusion of a point in the intersection goes to a point in each piece, the map $\mathbb{Z}\to \mathbb{Z}\oplus \mathbb{Z}$ is $1\mapsto (1,1)$ which is injective.  Hence it has kernel isomorphic to $0$.
Clearly \begin{eqnarray*} H_2(K\#\mathbb{R}P^2)&=&{\rm ker} f,\\
H_1(K\#\mathbb{R}P^2)&=&{\rm coker} f,\end{eqnarray*}
so once we know $f$ we have the homology groups.
From the following diagrams, we can see the generator $[t]$ of $H_1$ of the intersection, maps to twice the generator $[c]$ of $H_1(\mathbb{R}P^2 \backslash D^2)$ and twice one of the generators  $[a]$ of $H_1(K \backslash D^2)$:

$$[t]\mapsto [a]-[b]+[a]-[b]=2[a],\qquad [t]\mapsto [c]+[c]=2[c]$$
Thus \begin{eqnarray*} H_2(K\#\mathbb{R}P^2)&=&{\rm ker} f=0,\\
H_1(K\#\mathbb{R}P^2)&=&{\rm coker} f=\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z},\\ 
H_0(K\#\mathbb{R}P^2)&=&\mathbb{Z}.
\end{eqnarray*}
