List of common homology groups

Surface-like

Space		Homology groups	[ilmath]\Delta[/ilmath]-complex	Components	Comment
[ilmath]\mathbb{S}^2[/ilmath]	2-sphere, in [ilmath]\mathbb{R}^3[/ilmath]	[ilmath]H_0(\mathbb{S}^2)\cong\mathbb{Z} [/ilmath] [ilmath]H_1(\mathbb{S}^2)\cong 0[/ilmath] (i.e.: trivial group) [ilmath]H_2(\mathbb{S}^2)\cong\mathbb{Z} [/ilmath] [ilmath]H_k(\mathbb{S}^2)\cong 0[/ilmath] for [ilmath]k\ge 3[/ilmath]	[ilmath]\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}[/ilmath]