Let $F$ be a vector field defined on $\mathbb{S}^2$ such that $$F(x,y, z)-[ F(x,y, z)\cdot k ]k=F(-x,-y, z)-[F(-x,-y, z)\cdot k]k$$ and $F(x,y,z)=F(-x,-y,-z)$. Here $k=(0,0,1)$.
Indeed $F=(f_1,f_2, f_3)$ such that $f_1, f_2$ are symmetric about $xy-$plane and $f_3$ is anti symmetric about $xy-$ plane.
Suppose $F$ is continuous everywhere except $(0,0, \pm1)$. What can we say about the index of $F$ near $(0,0, \pm1)$? Can we conclude that they are even numbers? Or can they be both equal to 1?
I am just learning about index of maps, and I understand that this might be a trivial question.
I assume that by vector field you mean a vector field everywhere tangent to the sphere. The tangent space at the poles is horizontal, so you should just think about a vector field $g$ on the plane with $g(x,y)=g(-x,-y)$. I don’t know your definition of index. For example, any regular value of $g/|g|$ mapping circle to circle will have an even number of preimages. Or, consider the vector field on a small circle centered at the origin; it starts and ends at the same place at any pair of opposite points, and so it makes twice that number of turns as we go entirely around the circle.