# Intersection number via tangent spaces

Assume that finite groups $$G_1$$ and $$G_2$$ act smoothly on a manifold $$M$$ in such a way that the fixed point set, $$M^{G_1\cap G_2}$$, is an oriented closed manifold, $$M^{G_1}$$ and $$M^{G_2}$$ are its oriented closed submanifolds (they are orientable and we fix their orientations) and $$M^{G_1}$$ and $$M^{G_2}$$ are transverse in $$M^{G_1\cap G_2}$$ (all the assumptions are made so that we have a well-defined intersection number of $$M^{G_1}$$ and $$M^{G_2}$$ in $$M^{G_1\cap G_2}$$).

Assume that $$M^{G_1}\cap M^{G_2}$$ is a two-point set - $$M^{G_1}\cap M^{G_2}=\{x_1,x_2\}$$ and consider tangent spaces $$T_{x_1}M^{G_1}$$, $$T_{x_1}M^{G_2}$$ and $$T_{x_1}M^{G_1\cap G_2}$$. They are endowed with an $$\mathbb{R}G_1$$, $$\mathbb{R}G_2$$ and $$\mathbb{R}(G_1\cap G_2)$$-module structures respectively (analogously for the tangent spaces at $$x_2$$).

Is there a way to compute the intersection number of $$M^{G_1}$$ and $$M^{G_2}$$ in $$M^{G_1\cap G_2}$$ basing only on the $$\mathbb{R}G$$-module structures of the tangent spaces mentioned before (here $$G\in\{G_1,G_2,G_1\cap G_2\}$$)?

• Consider a simple example. Let $M$ be the sphere embedded in $\mathbb{R}^3$, $G_1$ the $2$-element group generated by reflection at the $x,y$-plane and $G_2$ the $2$-element group generated by reflection at the $x,z$-plane. Then $x_{1/2}=(\pm 1, 0,0)$. The intersection numbers are $1$ and $-1$ due to a switch in orientation. How do you see this orientation switch in the module structure? – quarague Feb 5 at 12:17