I found the Cross-cap function in $\mathbb{R}^3$ as follows: $$f(x,y,z)=(yz,2xy,x^2-y^2),$$
My questions are (I couldn't show any progress for Q1,2.I have thought hard but had no clue):
Q1: Is there a form for crosscap in $\mathbb{R}^4$?
Q2: Is there a way to show this is proper (the preimage of a compact space is compact)?
Q3: Is there a way to show this is injective?
All my attempts are for linear functions, so none of them work.
I tried to show $f(x,y,z) = 0 \Rightarrow x = y = z = 0$. But I get $x=y=0$, with no constraint on $z$.
I tried to show $f(x_1,y_1,z_1) = f(x_2,y_2,z_2) \Rightarrow (x_1,y_1,z_1) = (x_2,y_2,z_2)$ but made no progress.
I tried to solve $f(x,y,z) = (a,b,c,d)$ for $x,y,z$ on mathematica, and the result is scary.