Fundamental groups and integral homology of spheres, tori, projective spaces, closed surfaces, and wedges of spheres.
| \(X\) | \(\pi_1\) | \(H_*\) | Notes |
|---|---|---|---|
| \(S^n\), \(n\geq 2\) | \(1\) | \(\ZZ\) in degrees \(0, n\) | simply connected, its own universal cover |
| \(S^1\) | \(\ZZ\) | \(\ZZ, \ZZ\) | universal cover \(\RR\) |
| \(T^n\) | \(\ZZ^n\) | \(\ZZ^{\binom nk}\) in degree \(k\) | universal cover \(\RR^n\); product of circles |
| \(\RP^n\), \(n\geq 2\) | \(\ZZ/2\) | \(\ZZ, \ZZ/2, 0, \ZZ/2, \dots\) | universal cover \(S^n\); \(H_n = \ZZ\) iff \(n\) odd |
| \(\CP^n\) | \(1\) | \(\ZZ\) in every even degree \(\leq 2n\) | no odd cells, so all boundary maps vanish |
| Klein bottle \(K\) | \(\gens{a,b \st aba\inverseof{b}}\) | \(\ZZ, \ZZ\oplus\ZZ/2, 0\) | non-orientable, so \(H_2 = 0\) |
| \(\Sigma_g\), genus \(g\) | \(\gens{a_i,b_i \st \prod[a_i,b_i]}\) | \(\ZZ, \ZZ^{2g}, \ZZ\) | closed orientable |
| \(N_k\), \(k\) crosscaps | \(\gens{a_i \st \prod a_i^2}\) | \(\ZZ, \ZZ^{k-1}\oplus\ZZ/2, 0\) | closed non-orientable |
| \(\bigvee_n S^1\) | free on \(n\) | \(\ZZ, \ZZ^n\) | \(\pi_1\) free, homology free |
| \(\bigvee_n S^2\) | \(1\) | \(\ZZ, 0, \ZZ^n\) | |
| Möbius band | \(\ZZ\) | \(\ZZ, \ZZ\) | deformation retracts to its core circle |
| \(S^1\times S^2\) | \(\ZZ\) | \(\ZZ, \ZZ, \ZZ, \ZZ\) |
Constructions
Wedges. For good pointed spaces, \(\tilde H_*(X\vee Y) \cong \tilde H_*(X)\oplus\tilde H_*(Y)\), and if \(X\) and \(Y\) are CW complexes wedged at \(0\)-cells, \(\pi_1(X\vee Y)\cong\pi_1(X)*\pi_1(Y)\) by van Kampen’s theorem.
Products. \(\pi_1(X\times Y)\cong\pi_1(X)\times\pi_1(Y)\), and by the Künneth theorem for CW complexes \[ H_n(X\times Y)\cong \bigoplus_{i+j=n} H_i(X)\tensor H_j(Y)\ \oplus \bigoplus_{i+j=n-1}\Tor(H_i(X),H_j(Y)). \]
Deleting a point. If a closed \(n\)-manifold \(M\) has a CW structure with a single \(n\)-cell, then \(M\) minus a point in that cell deformation retracts onto the \((n-1)\)-skeleton; a closed surface minus a point deformation retracts onto a wedge of circles.
Connected sums. For closed \(n\)-manifolds with \(n\geq 2\), the homology of \(M\# N\) is computed by the Mayer–Vietoris sequence for the decomposition along the separating \(S^{n-1}\); \(\Sigma_g\#\Sigma_h\cong\Sigma_{g+h}\).
Spaces distinguished by one invariant
($T^2$ and the Klein bottle)
\(H_1(T^2)\cong\ZZ^2\) and \(H_1(K)\cong\ZZ\oplus\ZZ/2\), and \(H_2(T^2)\cong\ZZ\) while \(H_2(K)=0\).
($\RP^2$ and $S^2$)
\(\pi_1(\RP^2)\cong\ZZ/2\) and \(\pi_1(S^2)=1\).
($\RP^3$ and $S^1\times S^2$)
\(\pi_1(\RP^3)\cong\ZZ/2\) and \(\pi_1(S^1\times S^2)\cong\ZZ\); also \(H_1(\RP^3)\cong\ZZ/2\) and \(H_1(S^1\times S^2)\cong\ZZ\).
($S^1\vee S^1\vee S^2$ and $T^2$)
Both have homology \(\ZZ,\ZZ^2,\ZZ\), and \(\pi_1(S^1\vee S^1\vee S^2)\) is free of rank \(2\) while \(\pi_1(T^2)\cong\ZZ^2\) is abelian, so they are not homotopy equivalent.