nLab semialgebraic manifold

Idea

A semi-algebraic set is much like a variety over the real numbers, a subset of a Cartesian space $\mathbb{R}^n$ being the joint vanishing locus in of a finite set of polynomials intersected with inequalities.

A semi-algebraic manifold is much like a corresponding scheme, a space that is locally like a semi-algebraic set.

Examples

Example

The Fulton-MacPherson compactifications $FM_n\left( \mathbb{R}^d\right)$ of ordered configuration spaces of points are semi-algebraic manifolds (Lambrechts-Volic 14, Prop. 5.2).

Moreover, the canonical forgetful functions $FM_{n + k}\left( \mathbb{R}^d\right) \longrightarrow FM_{n}\left( \mathbb{R}^d\right)$ are semi-algebraic fiber bundles (Lambrechts-Volic 14, Theorem 5.8).

References

• J. Bochnak, M. Coste, and M.-F. Roy. Geometrie algebrique reelle, volume 12 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1987

A theory of differential forms on semi-algebraic manifolds was sketched in Appendix 8 of

• Maxim Kontsevich, Yan Soibelman, Deformations of algebras over operads and the Deligne conjecture. In Conference Moshe Flato 1999, Vol. I (Dijon) , volume 21 of Math. Phys. Stud., pages 255–307. Kluwer Acad. Publ., Dordrecht, 2000 (arXiv:math/0001151)

Details are provided, the de Rham complex of semi-algebraic differential forms on a semi-algebraic manifold is constructed in and their real homotopy theory is studied in

• Robert Hardt, Pascal Lambrechts, Victor Tourtchine, Ismar Volić, Real Homotopy Theory of Semi-Algebraic Sets, Algebr. Geom. Topol. 11 (2011) 2477-2545 (arXiv:0806.0476)

Discussion of the example of Fulton-MacPherson compactifications of configuration spaces of points includes

• Pascal Lambrechts, Ismar Volic, section 5 of Formality of the little N-disks operad, Memoirs of the American Mathematical Society ; no. 1079, 2014 (arXiv:0808.0457, doi:10.1090/memo/1079)

See also

• Wikipedia, Semialgebraic set