Math 2.0

Several complex variables and analytic spaces Local analytic geometry Theorem

Triangulability of analytic sets

Builds on Subanalytic setSimplicial complexHomeomorphismTriangulationSemianalytic setComplex analytic variety

Statement

Let X⊆ℝnX \subseteq \R^n be a closed subanalytic set and let A1,…,AkA_1, \dots, A_k be subanalytic subsets of XX. Then there are a locally finite simplicial complex KK and a homeomorphism h ⁣:∣K∣→Xh \colon \abs{K} \to X from the polyhedron of KK onto XX with the following property. Each AjA_j is a union of sets h(σ∘)h(\sigma^\circ), where σ∘\sigma^\circ is the interior of a simplex σ\sigma of KK.

Remarks

Łojasiewicz proved the statement for semianalytic sets, and Hironaka extended it to subanalytic sets. The proof first decomposes XX into finitely many pieces on which suitable coordinate projections are finite and well behaved, and then triangulates by induction on nn. Every closed real analytic subset of ℝn\R^n is semianalytic, and every complex analytic set in ℂn=ℝ2n\C^n = \R^{2n} is a closed real analytic subset. So both kinds of analytic sets are triangulable, and a triangulation can be chosen compatible with the singular locus. For instance, the cone {x2+y2=z2}\set{x^2 + y^2 = z^2} in ℝ3\R^3 can be triangulated with the vertex at the origin as a vertex of KK.

Going deeper

For a student

Triangulability implies that a compact analytic set has finitely generated homology and a well defined Euler characteristic. It also implies local conic structure: a small neighbourhood of a point of an analytic set is homeomorphic to the cone over its link. The choice of a triangulation compatible with the subsets AjA_j lets one treat a set together with a stratification of it.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Complex analytic variety, concept, no evidence levelComplex analytic vari…conceptHomeomorphism, concept, no evidence levelHomeomorphismconceptSemianalytic set, concept, no evidence levelSemianalytic setconceptSimplicial complex, concept, no evidence levelSimplicial complexconceptSubanalytic set, concept, no evidence levelSubanalytic setconceptTriangulation, concept, no evidence levelTriangulationconceptTriangulability of analytic sets, statement, no evidence levelTriangulability of an…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex analytic variety concept
  2. Homeomorphism concept
  3. Semianalytic set concept
  4. Simplicial complex concept
  5. Subanalytic set concept
  6. Triangulation concept
  7. Triangulability of analytic sets statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Subanalytic_set
About this page
Tag6SZDXD3KYH
URIm2://6SZDXD3KYH
Kindstatement, theorem
WritingW0 stub
MSC32B25, 32B20, 57Q15
StatusThis copy of the site has no build data. Run m2 status 6SZDXD3KYH.
AboutSubanalytic set, Triangulation
UsesSubanalytic set, Simplicial complex, Homeomorphism, Triangulation, Semianalytic set, Complex analytic variety

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.