How to classify continuous self maps on $L(2,1)\# L(3,1)$ up to homotopy? Here $L(2,1)$ and $L(3,1)$ are lens spaces.
The reason for considering this manifold is that its fundamental group is $\mathbb{Z}_{2}\ast \mathbb{Z}_{3}$ which is isomorphic to ${\rm PGL}(2,\mathbb{Z})$, so that its elements can be written as matrices.
I am also interested in the homotopy classification of maps between any two closed oriented 3-manifolds.
Thanks to everyone.