Antipodal map

The antipodal map $x\mapsto -x$ on $S^m$ has degree $(-1{)}^{m+1}$.

If a continuous map $f:S^d\to S^d$ has no fixed point, the formula $H_t(x)=((1-t)f(x)-tx)/\Vert (1-t)f(x)-tx\Vert$ is a homotopy from $f$ to the antipodal map. The denominator could vanish only at $t=1/2$ and $f(x)=x$, which is excluded. The homotopy invariance of mapping degree therefore gives $\mathrm{deg}f=(-1{)}^{d+1}$.

An odd map between spheres is a continuous map $f:S^n\to S^m$ satisfying $f(-x)=-f(x)$. Thus it is equivariant for the two antipodal maps. It descends to a continuous map of Real projective spaces.

Suppose a continuous map $g:S^n\times S^n\to S^n$ changes sign on negating either input. Its quotient $\bar{g}:{\mathbb{RP}}^n\times {\mathbb{RP}}^n\to {\mathbb{RP}}^n$ pulls back the real tautological line bundle to the tensor product of vector bundles $p_1^{\ast }{\gamma }_n\otimes p_2^{\ast }{\gamma }_n$. The fiber map sends $(sx)\otimes (ty)$ to $stg(x,y)$. The First Stiefel–Whitney class of a tensor product of real line bundles consequently gives ${\bar{g}}^{\ast }a=a_1+a_2$. The Künneth theorem and the mod-two cohomology ring of real projective space implyEvery interior binomial coefficient in row $n+1$ must be even. By binomial coefficients with even interior terms, $n=2^k-1$ for some $k\ge 0$. This is a necessary condition, not an assertion of existence in every such dimension.

For an odd map between spheres $f:S^n\to S^m$ and its quotient $\bar{f}$, there is an isomorphism ${\gamma }_n\cong {\bar{f}}^{\ast }{\gamma }_m$ of real tautological line bundles. With $x$ a unit vector, send $tx$ to $tf(x)$; replacing $x$ by $-x$ and $t$ by $-t$ gives the same vector. This proves both linearity on fibers and well-definedness. Therefore ${\bar{f}}^{\ast }{w}_1({\gamma }_m)=w_1({\gamma }_n)$. The mod-two cohomology ring of real projective space then implies $n\le m$, since a vanishing $(m+1)$st power must pull back to a vanishing power.

Let a finite group $G$ act smoothly on a smooth manifold, and suppose that a $G$-invariant differential form $\omega$ is an exact differential form, say $\omega =d\eta$. Averaging gives the invariant primitiveIf the action is free, invariant forms descend uniquely through the resulting covering map.

For the double covering $\pi :S^{2n}\to {\mathbb{RP}}^{2n}$ and the antipodal map $a$, every pulled-back top form satisfies $a^{\ast }{\pi }^{\ast }\omega ={\pi }^{\ast }\omega$. Since $\mathrm{deg}a=-1$, its integral over the sphere is its own negative and hence vanishes. It is therefore exact on the sphere. Averaging a primitive under $a$ and descending it proves that $\omega$ is exact on real projective space.