6520-2023-05

Prove that if \(X\) and \(Y\) are path connected spaces then

Let \(\Sigma _2\) be the closed, orientable genus two surface. Show that \(\pi _1(\Sigma _2,x)\) has a normal subgroup \(G\) with the quotient group \(\pi _1(\Sigma _2)/G\) isomor- phic to \(\mathbb {Z}\). You can use a picture as part of your proof.

Let \(\Delta ^n\) be the \(n\)-simplex with its usual \(\Delta \)-complex structure. Find all subcomplexes \(X\) of \(\Delta ^n\) such that \(\tilde {H}_{n-1}(X)\) is non-zero.

Let \(X=S^1\times [0,1]\) and \(\partial X=S^1\times \{0,1\}\). Calculate \(H_i(X,\partial X)\) for all \(i\).

1. Construct a single chain complex \(C\) such that \(C\) has the following three properties:

   • \(H_i(C)\) is non-zero if and only \(i=1\).
   • There is a coefficient group \(G_0\) such that \(\tilde {H}^i(C;G_0)=0\) for all \(i\).
   • There is a coefficient group \(G_1\) such that \(\tilde {H}^i(C;G_1)\) is non- zero if and only if \(i=1,2\).
2. Find a topological space \(X\) such that \(H_i(C)=\tilde {H}_i(X)\) for all \(i\).

Show that \(S^2\vee S^4\) is not homotopy equivalent to a compact manifold without boundary.