= Odd maps pull back the real tautological line bundle
{title2=$\gamma_n\cong\bar f^*\gamma_m$}
For an <odd map between spheres> $f:S^n\to S^m$ and its quotient $\bar f$, there is an <isomorphism> $\gamma_n\cong\bar f^*\gamma_m$ of <real tautological line bundles>. With $x$ a unit vector, send $t x$ to $t f(x)$; replacing $x$ by $-x$ and $t$ by $-t$ gives the same vector. This proves both linearity on fibers and well-definedness. Therefore $\bar f^*w_1(\gamma_m)=w_1(\gamma_n)$. The <mod-two cohomology ring of real projective space> then implies $n\leq m$, since a vanishing $(m+1)$st power must pull back to a vanishing power.
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