List of common homology groups

Surface-like

Space		Homology groups	$\Delta$-complex	Components	Comment
$\mathbb{S}^2$	2-sphere, in $\mathbb{R}^3$	$H_0(\mathbb{S}^2)\cong\mathbb{Z}$ $H_1(\mathbb{S}^2)\cong 0$ (i.e.: trivial group) $H_2(\mathbb{S}^2)\cong\mathbb{Z}$ $H_k(\mathbb{S}^2)\cong 0$ for $k\ge 3$	$\begin{xy}\xymatrix{\bullet \ar@{}@(l,u)[]_{v_0} \ar[r]^a \ar[d]_a & \bullet \ar@{}@(r,u)[]^{v_1} \\ \bullet \ar@{}@(l,d)[]^{v_1} \ar[ur]^A_B_(.75)c^(.75)c \ar@{<-}[r]_b & \bullet \ar[u]_b \ar@{}@(r,d)[]_{v_2} }\end{xy}$