nLab
semialgebraic manifold

Contents

Idea

A semi-algebraic set is much like a variety over the real numbers, a subset of a Cartesian space n\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( d)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( d)FM n( d)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

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

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

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

See also