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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 $G\in\{G_1,G_2,G_1\cap G_2\}$)?

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    $\begingroup$ 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? $\endgroup$ – quarague Feb 5 at 12:17

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