A table of homology and cohomology groups Does anyone know where I can find a table of the homology and cohomology groups, with different coefficients, of standard spaces - $S^1\times S^1$, Klein bottle, projective space, etc.?
 A: One place to look at is Topospaces (The Topology Wiki), which was suggested by a now-deleted user in a now-deleted answer. The description says

This is a pre-alpha stage topology wiki primarily managed by Vipul Naik, a Ph.D. in Mathematics at the University of Chicago. We have over 400 articles including some material in basic point-set topology. 

As the description suggests, the form is very preliminary, but there are useful articles such as Homology of complex projective space (with various coefficients) and several others. 
A: Here are some (integral) homology computations.

*

*Sphere ($S^n$, $n > 0$):  $H_0 = \mathbf{Z}$, $H_{0<k<n} = 0$, $H_n = \mathbf{Z}$, $H_{>n} = 0$.

*Torus ($S^1\times S^1$):  $H_0 = \mathbf{Z}$, $H_1 = \mathbf{Z}^2$, $H_2 = \mathbf{Z}$.

*Klein bottle:  $H_0 = \mathbf{Z}$, $H_1 = \mathbf{Z} \oplus \mathbf{Z}_2$, $H_{\ge 2} = 0$.

*Real projective space ($\mathbf{R}\mathbf{P}^n$):  $H_0 = \mathbf{Z}$, $H_{0 < 2k-1 < n} = \mathbf{Z}_2$, $H_{0<2k<n} = 0$, $H_{n\ \text{odd}} = \mathbf{Z}$, $H_{n\ \text{even}} = 0$, $H_{>n} = 0$.

*Complex projective space ($\mathbf{C}\mathbf{P}^n$):  $H_{0 \le 2k \leq 2n} = \mathbf{Z}$, $H_{0 < 2k+1 < 2n} = 0$, $H_{>2n} = 0$.

*Quaternionic projective space ($\mathbf{HP}^n$):  $H_{0\leq 4k\leq 4n} = \mathbf{Z}$, $H_{0< 4k+1, 4k+2, 4k+3 < 4n} = 0, H_{>4n} = 0$

*Octonionic projective plane ($\mathbf{OP}^2$): $H_0 = \mathbf{Z}$, $H_8 = \mathbf{Z}$, $H_{16} = \mathbf{Z},$ $H_{0<k<8} = H_{8<k<16} = H_{>n} = 0$

*Möbius band ($\mathscr{M}$): $H_0= \mathbf{Z}, H_1 = \mathbf{Z}, H_{\geq 2} = 0$
I've made this answer "community wiki", so feel free to add anything I've missed.
