Continuous Map on Compact Oriented Manifold of Higher Degree

Question

Let $M$ be a compact oriented manifold of dimension $\ge 3$. Is there any known obstructions of $M$ for it to admit a continuous self-map of degree $>1$?

Note: the case when $\dim(M) = 2$ is known (see this post).

Remark: In (the top answer of) the post above, it was explained that if $M$ has positive simplicial volume (denoted as $||M||$), then it does not admit a self-map of degree $>1$. However, if $M$ is simply-connected, we have $||M|| = 0$. So 2 follow-up questions are:

1. Let $M$ be a simply-connected compact oriented manifold of dimension $\ge 3$. Is there any known obstructions of $M$ for it to admit a continuous self-map of degree $>1$?
2. Original question, replace "obstruction" with "obstruction other than simplicial-volume".

Answer 1

This was studied quite a bit in the 3-dimensional case. The main obstruction is positive simplicial volume, e.g. hyperbolic manifolds do not admit such maps. For details see

Sun, Hongbin; Wang, Shicheng; Wu, Jianchun; Zheng, Hao, Self-mapping degrees of 3-manifolds, Osaka J. Math. 49, No. 1, 247-269 (2012). ZBL1241.55002.