Homological stability for unordered configuration spaces
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Homological stability for unordered configuration spaces. / Randal-Williams, Oscar.
I: Quarterly Journal of Mathematics, Bind 64, Nr. 1, 2013.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Homological stability for unordered configuration spaces
AU - Randal-Williams, Oscar
PY - 2013
Y1 - 2013
N2 - 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.
AB - 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.
KW - math.AT
KW - 55R40, 55R80
M3 - Journal article
VL - 64
JO - Quarterly Journal of Mathematics
JF - Quarterly Journal of Mathematics
SN - 0033-5606
IS - 1
ER -
ID: 37731365