Let $M^n_{\mathbb{R}}$ be a smooth manifold that is the set of real points of a smooth complex manifold $M^n_{\mathbb{C}}$ then we have: $$\sum_{j=0}^n b_j(M^n_{\mathbb{R}})\leq \sum_{j=0}^{2n}b_j(M^n_{\mathbb{C}}).$$ Where $b_j$ is the $j$-th Betti number with $\mathbb{Z}/2$-coefficients. We consider $M^n_{\mathbb{R}}$ as the set of fixed points of the involution given by by complex conjugation on $M^n_{\mathbb{C}}$.

A modern proof of this inegality can be found in: A. Borel. "Seminar on Transformation groups, (with contributions by G. Bredon, E.E. Floyd, D. Montgomery, R. Palais)" Ann. of M. Studies n. 46, Princeton UniversityPress, 1960.

It uses spectral sequences for the $\mathbb{Z}/2$-equivariant cohomology.

Let me give an example of the use of Smith's theory in real algebraic geometry. Smith theory gives a way to compute fixed point sets in terms of equivariant cohomology, for a "modern" treatment see: Dwyer, William G.; Wilkerson, Clarence W. "Smith theory revisited." Ann. of Math. (2) 127 (1988), no. 1, 191–198.

Let $M^n_{\mathbb{R}}$ be a smooth manifold that is the set of real points of a smooth complex manifold $M^n_{\mathbb{C}}$ then we have: $$\sum_{j=0}^n b_j(M^n_{\mathbb{R}})\leq \sum_{j=0}^{2n}b_j(M^n_{\mathbb{C}}).$$ Where $b_j$ is the $j$-th Betti number with $\mathbb{Z}/2$-coefficients. We consider $M^n_{\mathbb{R}}$ as the set of fixed points of the involution given by by complex conjugation on $M^n_{\mathbb{C}}$.

A modern proof of this inegality can be found in: A. Borel. "Seminar on Transformation groups, (with contributions by G. Bredon, E.E. Floyd, D. Montgomery, R. Palais)" Ann. of M. Studies n. 46, Princeton UniversityPress, 1960.

It uses spectral sequences for the $\mathbb{Z}/2$-equivariant cohomology.

Let $M^n_{\mathbb{R}}$ be a smooth manifold that is the set of real points of a smooth complex manifold $M^n_{\mathbb{C}}$ then we have: $$\sum_{j=0}^n b_j(M^n_{\mathbb{R}})\leq \sum_{j=0}^{2n}b_j(M^n_{\mathbb{C}}).$$ Where $b_j$ is the $j$-th Betti number with $\mathbb{Z}/2$-coefficients.

A modern proof of this inegality can be found in: A. Borel. "Seminar on Transformation groups, (with contributions by G. Bredon, E.E. Floyd, D. Montgomery, R. Palais)" Ann. of M. Studies n. 46, Princeton UniversityPress, 1960.

It uses spectral sequences for the $\mathbb{Z}/2$-equivariant cohomology.

Let $M^n_{\mathbb{R}}$ be a smooth manifold that is the set of real points of a smooth complex manifold $M^n_{\mathbb{C}}$ then we have: $$\sum_{j=0}^n b_j(M^n_{\mathbb{R}})\leq \sum_{j=0}^{2n}b_j(M^n_{\mathbb{C}}).$$ Where $b_j$ is the $j$-th Betti number with $\mathbb{Z}/2$-coefficients. We consider $M^n_{\mathbb{R}}$ as the set of fixed points of the involution given by by complex conjugation on $M^n_{\mathbb{C}}$.

A modern proof of this inegality can be found in: A. Borel. "Seminar on Transformation groups, (with contributions by G. Bredon, E.E. Floyd, D. Montgomery, R. Palais)" Ann. of M. Studies n. 46, Princeton UniversityPress, 1960.

It uses spectral sequences for the $\mathbb{Z}/2$-equivariant cohomology.