Homology Group of some Common Spaces

Homology of Circle
\displaystyle H_n(S^1)=\begin{cases}\mathbb{Z}&\text{for}\ n=0,1\\  0&\text{for}\ n\geq 2.  \end{cases}

Homology of Torus
\displaystyle H_n(T)=\begin{cases}\mathbb{Z}\oplus\mathbb{Z}&\text{for}\ n=1\\  \mathbb{Z}&\text{for}\ n=0, 2\\  0&\text{for}\ n\geq 3.  \end{cases}

Homology of Real Projective Plane
\displaystyle H_n(\mathbb{R}P^2)=\begin{cases}  \mathbb{Z}&\text{for}\ n=0\\  \mathbb{Z}_2&\text{for}\ n=1\\  0&\text{for}\ n\geq 2.  \end{cases}

Homology of Klein Bottle
\displaystyle H_n(K)=\begin{cases}  \mathbb{Z}&\text{for}\ n=0\\  \mathbb{Z}_2\oplus\mathbb{Z}&\text{for}\ n=1\\  0&\text{for}\ n\geq 2.  \end{cases}

Also see How to calculate Homology Groups (Klein Bottle).

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