Manifold-theoretic compactifications of configuration spaces

By Dev Sinha. 2003

134 citations recorded by the source corpus.

Abstract.We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to Fulton–MacPherson and Axelrod–Singer in the setting of smooth manifolds, as well as a simplicial variant of this compactification initiated by Kontsevich. Our constructions are elementary and give simple global coordinates for the compactified configuration space of a general manifold embedded in Euclidean space. We stratify the canonical compactification, identifying the difieomorphism types of the strata in terms of spaces of configurations in the tangent bundle, and give completely explicit local coordinates around the strata as needed to define a manifold with corners. We analyze the quotient map from the canonical to the simplicial compactification, showing it is a homotopy equivalence. Using global coordinates we define projection maps and diagonal maps, which for the simplicial variant satisfy cosimplicial identities.

Subfields: Algebra and Number Theory, Mathematical Physics, Pathology and Forensic Medicine

View Paper: https://www.semanticscholar.org/paper/4124b23f78226c99b265f2325ea8a07d5fe8d5dd

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In Algebra and Number Theory

1. Ismar Volic
2. P. Lambrechts
3. Michael Weiss
4. Pedro Boavida de Brito
5. Dev Sinha
6. Victor Tourtchine
7. Victor Turchin
8. Ryan Budney

In Mathematical Physics

1. Dev Sinha
2. Ismar Volic
3. P. Lambrechts
4. Pedro Boavida de Brito
5. Michael Weiss
6. Victor Tourtchine
7. Victor Turchin
8. G. Arone

In Pathology and Forensic Medicine

1. Ismar Volic
2. P. Lambrechts
3. Michael Weiss
4. Pedro Boavida de Brito

Citations in the corpus

• Formality of the Little N-Disks Operad (2008, methodology): “We follow Kontsevich’s approach which was developed by Sinha in [18].”
• Operads and knot spaces (2004, background): “Let K m denote the m th Kontsevich operad, introduced in [22], whose entries are compactified configuration spaces and which is weakly equivalent to the little m -disks operad [37]; see Definition 4.1 and Theorem 4.5.”
• Spaces of smooth embeddings and configuration categories (2015, background): “Sinha [34] has developed a rather nice way to represent elements of M(k,Φ) by pictures.”