HOMOLOGICAL STABILITY FOR UNORDERED CONFIGURATION SPACES
By O. Randal-Williams. 2011
87 citations recorded by the source corpus.
This paper consists of two related parts. In the first part we give a self-contained proof of homological stability for the spaces C_n(M;X) of configurations of n unordered points in a connected open manifold M with labels in a path-connected space X, with the best possible integral stability range of 2* \leq n. Along the way we give a new proof of the high connectivity of the complex of injective words. If the manifold has dimension at least three, we show that in rational homology the stability range may be improved to * \leq n. In the second part we study to what extent the homology of the spaces C_n(M) can be considered stable when M is a closed manifold. In this case there are no stabilisation maps, but one may still ask if the dimensions of the homology groups over some field stabilise with n. We prove that this is true when M is odd-dimensional, or when the field is F_2 or Q. It is known to be false in the remaining cases.
Subfields: Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/bc093ffae604a3d6754d01715e2dfcf42d020b31
How researchers see this paper
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In Computational Theory and Mathematics
- Benson Farb
- O. Randal-Williams
- Nathalie Wahl
- Thomas Church
- Jenny Wilson
- Samuel J. Taylor
- Rohit Nagpal
- Wen-yuan Yang
In Mathematical Physics
- O. Randal-Williams
- Nathalie Wahl
- Benson Farb
- Andrew Putman
- Jenny Wilson
- Steven V. Sam
- Michael Weiss
- Rohit Nagpal
Citations in the corpus
- Higher-order representation stability and ordered configuration spaces of manifolds (2016, unknown): “An augmented semisimplicial space A (cid:15) gives rise to a homology spectral sequence; see for example Randal-Williams [35, Section 2.3].”
- Representation Stability (2014, background): “The claim on dimk(Vn) is obtained by noting that dimk(Vn) = χVn(Id) = P (n, 0, . . . , 0).”
- Homological stability for automorphism groups (2014, background): “If C satisfies LH3 at (A,X) with slope k = 1, then the isomorphism range can be improved to be equal to the epimorphism range i ≤ n2 by a variation of the argument below (see [67, 37] for the case of the symmetric groups).”