Let $M$ be a compact oriented smooth $n$-manifold, with $H_{dR}^1(M)=0$. For which integers $k$ does there exist a smooth map $f : M → T^n$ of degree $k$?
I know that if $M$ is simply-connected, we can lift the map to the universal cover, and then conclude that $k=0$.
However, not every $M$ with $H_{dR}^1(M)=0$ is simply-connected. Is there any other method to solve this problem?
On
The de Rham cohomology ring of the $n$-dimensional torus is the Grassmann algebra $\bigwedge\mathbf Z^n$ and is generated by elements of degree $1$. If $f\colon M\rightarrow T^n$ is a differentiable map, then the induced morphism of algebras $$ f^\star\colon H_{dR}^\star(T^n)\rightarrow H_{dR}^\star(M) $$ is zero in degree $1$ since $H_{dR}^1(M)=0$ by assumption. It follows that $f^\star$ is zero in degree $n$ as well. Hence, the degree of $f$ is zero too.
You can still lift through the universal cover, because the image of any homomorphism $\pi_1(M) \to \Bbb Z^n$. It factors through $H_1(M)$ since $\Bbb Z^n$ is abelian, and kills the torsion subgroup of $H_1(M)$ because $\Bbb Z^n$ is torsion free. $H_1(M)/\text{Tors}(M) \cong \Bbb Z^{b_1}$, and your assumption is precisely that $b_1=0$.