All manifolds will be assumed to be closed, oriented, and connected.


Let $f\colon M\to M$ be a map of degree $\pm 1$. It is not hard to show that $\pi\_1(f)$ is surjective.



> 
> What is an example of a *non* $\pi\_1$-injective, degree one, self-map of a three-manifold?
> 
> 
> 


If $\dim(M) = 2$, then $\pi\_1(f)$ is injective because the fundamental group of any surface is residually finite, and any finitely generated residually finite group is Hopfian.


If $\dim(M) = 3$ and if $M$ is either a spherical, or hyperbolic, or Haken manifold, then again $\pi\_1(M)$ is Hopfian. Thus $\pi\_1(f)$ is injective.


A related conjecture is the [Hopf conjecture](https://en.wikipedia.org/wiki/Hopf_conjecture#Self-maps_of_degree_1).



Example of a non $\pi_1$-injective, degree one, self-map of a three-manifold
In the book [Three-manifold groups](https://arxiv.org/abs/1205.0202) we find a giant flow chart showing what is known to follow from the assumption that $N$ is an irreducible, compact, orientable three-manifold with empty or toroidal boundary (such that $\pi\_1(N)$ is neither finite nor solvable). As one particular consequence, such manifolds are Hopfian.


Manifolds with finite fundamental group are Hopfian, so those are dealt with. Three-manifolds with solvable fundamental group are toroidal, thus Haken, and so dealt with. Three-manifolds which are reducible have fundamental group being a free product. However, residually finiteness passes is preserved under taking free products. So reducible manifolds are dealt with. (See, for example, Hempel's paper [Residual finiteness for three-manifold groups](https://www.degruyter.com/document/doi/10.1515/9781400882083-018/html).)


Thus all closed, oriented, and connected three-manifolds have fundamental group being Hopfian.




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Further references: [a mathoverflow question](https://mathoverflow.net/questions/61661)


