Show that if \(X = S^2 {\textstyle\coprod}_{\operatorname{id}} S^2\) is a pushout along the equators, then \(H_n(X) = [{\mathbf{Z}}, 0, {\mathbf{Z}}^3, 0, \cdots]\).
Covering Spaces
Describe all connected covering spaces of \({\mathbf{RP}}^2 \vee {\mathbf{RP}}^2\).
Homology
Compute the homology of the Klein bottle using the Mayer-Vietoris sequence and a decomposition \(K = M {\textstyle\coprod}_f M\)
Use the Kunneth formula to compute \(H^*(S^2\times S^2; {\mathbf{Z}})\).
Known to be \([{\mathbf{Z}}, 0, {\mathbf{Z}}^2, 0, {\mathbf{Z}}, 0, 0, \cdots]\).
Compute \(H^*(S^2 \vee S^2 \vee S^4)\)
Known to be \([{\mathbf{Z}}, 0, {\mathbf{Z}}^2, 0, {\mathbf{Z}}, 0, 0, \cdots]\).
Show that \(\chi(\Sigma_g + \Sigma_h) = \chi(\Sigma_g) + \chi(\Sigma_h) - 2\).