Are all PL-branched coverings from $\mathbb{T}^4$ to $\mathbb{R}P^4$ null-homotopic?

Question

I can't solve one problem for the course on branched coverings.

Here it goes: I have to show that any PL-branched covering (simply a surjective PL-map with discrete fibers) from $\mathbb{T}^4$ (4-dimensional real torus) to $\mathbb{R}P^4$ has a degree (maximal number of points in a fiber (it is well-defined)) at least $8$.

I have a theorem (of Berstein and Edmonds, theorem 2.5) that states that any branched covering from $\mathbb{T}^4$ to $S^4$ has a degree of at least $4.$

My attempts so far: I can solve this problem for the case of the maps $f: \mathbb{T}^4\to\mathbb{R}P^4$ such that $f_*:\pi_1(\mathbb{T}^4) \to \pi_1(\mathbb{R}P^4)$ is zero. (Here I can lift the map to $S^4$ and then by multiplicativity of degree I obtain the result).

However, in more general case, I don't have the lifting map. And of course, there are maps $f: \mathbb{T}^4\to\mathbb{R}P^4$ such that $f_*:\pi_1(\mathbb{T}^4) \twoheadrightarrow \pi_1(\mathbb{R}P^4).$ Still, perhaps all those maps are not PL-branched coverings?

Answer 1

First of all, your definition of a (PL) branched covering is wrong. Check your lecture notes for the correct definition or/and talk to your professor.

The correct definition (for maps between compact PL manifolds) is that $f: X^n\to Y^n$ is a branched covering if

(1) there exists a (possibly empty) codimension 2 subcomplex $B\subset Y$ with $A:= f^{-1}(B)$ such that the restriction $f|_{X-A}$ is a covering map (to its image).

(2) In a regular neighborhood of every point $a\in A$ the map $f$ satisfies some further conditions, namely, it is the cone over a branched covering map between links.

It then follows that the preimage of every point in $Y$ is finite. The degree of $f$ (as a branched-covering) is then the cardinality of the preimage of (any) point $y\in Y-B$.

Remark. 1. By compactness, part (1) is equivalent to the requirement that $f|_{X-A}$ is a local PL homeomorphism.

2. For the purpose of your question, Part (1) will suffice.

3. Branched covering maps can be also defined for maps between noncompact manifolds and in the two other standard categories (TOP and DIFF).

Now, the hint for solving the problem is that you should first prove that if $X$ is orientable and $Y$ is non-orientable, and $f: X\to Y$ is a branched covering, then the image $f_*(\pi_1(X))$ is contained in the orientation-subgroup of $\pi_1(Y)$. (This is false for general surjective PL maps with finite fibers.) This is as much as I am willing to say, the rest is up to you to figure out.