I am running a qualifying exam prep course. A question I posed to my students was:
Suppose that $M$ and $N$ are compact, oriented manifolds and $f:M\longrightarrow N$ is of degree $1$. Show that $f_*:\pi_1(M)\longrightarrow \pi_1(N)$ is a surjection.
A student came up with the following:
There is some covering space, $p:E\longrightarrow N$, of $N$ that corresponds to the subgroup $f_*(\pi_1(M))\leq \pi_1(N)$. The lifting property gives us $\tilde{f}:M\longrightarrow E$ so that $p\circ\tilde{f}=f$. He then argued that $p$ has degree the number of sheets of the covering.
Their definition of degree is to take the preimage of a point $y\in N$ and determine for each $x\in f^{-1}(y)$ whether or not orientation is reversed. This leads to a problem in my student's solution: what if there are an infinite number of sheets in the covering? Then the degree is not really defined. If we had the homology definition or integral definition of degree, this is easily fixed. Is there an argument around this? If not, is there another solution using their definition?
Theorem. Let $M,N$ be closed,connected,oriented,smooth manifolds of the same dimension.Let $f:M\to N$ be a smooth map. Let $\:p:E\to N$ be the covering projection corresponding to the image of $\pi_1(M)$ under the homomorphism induced on fundamental groups by $f$, with $i$ sheets. Let $\:\tilde f:M\to E$ be the lifting of $f$; so $\:f=p\circ\tilde f$.
a) If $i$ is finite, then $\deg f$ is divisible by $i$.
b) If $i$ is infinite, then $\deg f=0$.
Proof without (co)homological tools. a) follows by the theorem on degree of composite maps.
b) Let $y\in N$ be a regular value of $f$. The set $\:Z=\tilde f(M)\cap p^{-1}(y)$ is finite as compact and discrete. Every point $z\in Z$ is a regular value of $\tilde f$ and $p$ is a local diffeomorphism concordant with orientations of $E$ and $N$. The manifold $E$ is not compact, so $\:0=\deg \tilde f =\deg (\tilde f ;z)$ for every $z\in Z$. The set $\:f^{-1}(y)$ is the disjoint union of $\:\tilde f^{-1}(z)$ for $z\in Z$, so $\:\deg f=0$ as the finite sum of $\deg(\tilde f;z)=0$ for $z\in Z$.