Math 6520 • Algebraic Topology

Let \(T^2\) be the real 2-dimensional torus and \(f:S^1\vee S^1\to T^2\) a continuous map. Prove or disprove the following
assertions:
\begin{enumerate}

\item
There is a continuous map \(g:T^2\to S^1\vee S^1\) such that \(f\circ g:T^2\to T^2\) is homotopy to the identity map.


\item
There is a continuous map \(g:T^2\to S^1\vee S^1\) such that \(g\circ f\) is homotopy to the identity map.

Justify your answers.

\end{enumerate}