Let $S$ be a compact algebraic surface with du Val singularities and $T$ be the minimal resolution of it. Is it true that $$ H_2(T,\mathbb{Z})=H_2(T,\mathbb{Z})\oplus(\oplus_{i}\mathbb{Z}E_i), $$ where $E_i$s are the exceptional curves of the resolution? Or under what condition does it hold?

The claim is true for the local case. That is, let $G \subset SL(2,\mathbb{Z})$ be a finite group and $X$ be the minimal resolution of $\mathbb{C}^2/G$. Then we have $$ H_2(X,\mathbb{Z})=\oplus_{i}\mathbb{Z}E_i $$ where $E_i$s are again the exceptional curves of the resolution. In other words, $E_i$s form a $\mathbb{Z}$-basis of $H_2(X,\mathbb{Z})$.

I think my problem is well-known and I would appreciate any help. I would also appreciate it if someone could teach me how other (co)homology groups change under minimal resolutions.

Let $S$ be a compact algebraic surface with du Val singularities and $T$ be the minimal resolution of it. Is it true that $$ H_2(T,\mathbb{Z})/(\oplus_{i}\mathbb{Z}E_i)\cong H_2(S,\mathbb{Z}), $$ where $E_i$s are the exceptional curves of the resolution? Or under what condition does it hold?

The claim is true for the local case. That is, let $G \subset SL(2,\mathbb{Z})$ be a finite group and $X$ be the minimal resolution of $\mathbb{C}^2/G$. Then we have $$ H_2(X,\mathbb{Z})=\oplus_{i}\mathbb{Z}E_i $$ where $E_i$s are again the exceptional curves of the resolution. In other words, $E_i$s form a $\mathbb{Z}$-basis of $H_2(X,\mathbb{Z})$.

I think my problem is well-known and I would appreciate any help. I would also appreciate it if someone could teach me how other (co)homology groups change under minimal resolutions.

Let $S$ be an algebraic surface with du Val singularities and $T$ be the minimal resolution of it. Is it true that $$ H_2(T,\mathbb{Z})=H_2(T,\mathbb{Z})\oplus(\oplus_{i}\mathbb{Z}E_i), $$ where $E_i$s are the exceptional curves of the resolution? This is true for the local case. That is, let $G \subset SL(2,\mathbb{Z})$ be a finite group and $X$ be the minimal resolution of $\mathbb{C}^2/G$. Then we have $$ H_2(X,\mathbb{Z})=\oplus_{i}\mathbb{Z}E_i $$ where $E_i$s are again the exceptional curves of the resolution.

I think my problem is well-known, and I would appreciate any help.

Let $S$ be a compact algebraic surface with du Val singularities and $T$ be the minimal resolution of it. Is it true that $$ H_2(T,\mathbb{Z})=H_2(T,\mathbb{Z})\oplus(\oplus_{i}\mathbb{Z}E_i), $$ where $E_i$s are the exceptional curves of the resolution? Or under what condition does it hold?

The claim is true for the local case. That is, let $G \subset SL(2,\mathbb{Z})$ be a finite group and $X$ be the minimal resolution of $\mathbb{C}^2/G$. Then we have $$ H_2(X,\mathbb{Z})=\oplus_{i}\mathbb{Z}E_i $$ where $E_i$s are again the exceptional curves of the resolution. In other words, $E_i$s form a $\mathbb{Z}$-basis of $H_2(X,\mathbb{Z})$.

I think my problem is well-known and I would appreciate any help. I would also appreciate it if someone could teach me how other (co)homology groups change under minimal resolutions.