Mapping Tori

Recall that the mapping torus of a map \(f:X\to X\) is the quotient space \(T_f\) obtained from \(X\times [0,1]\) by identifying \((x,1)\) with \((f(x),0)\) for every \(x\in X\). Compute the homology groups of \(T_f\) when \(f:S^n\to S^n\) is a map of degree \(d\) and \(n > 1\).

Let \(f:S^1\to S^1\) be given by \(f(z)=z^2\), where we regard \(S^1\) as the unit circle in \(\mathbb {C}\). Find a presentation of the fundamental group of the mapping torus \[ M_f=S^1\times I/(z,1)\sim (f(z),0) \] of \(f\).