Homological stability for unordered configuration spaces

TY - JOUR

T1 - Homological stability for unordered configuration spaces

AU - Randal-Williams, Oscar

PY - 2013/3

Y1 - 2013/3

N2 - This paper consists of two related parts. In the first partwegive a self-contained proof of homological stability for the spaces Cn (M; X) of configurations of n unordered points in a connected open manifoldM with labels in a path-connected space X, with the best possible integral stability range of 2* ≤ 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 3, we show that in rational homology the stability range may be improved to * ≤n. In the second part we study to what extent the homology of the spaces Cn (M) can be considered stable when M is a closed manifold. In this case there are no stabilization maps, but one may still ask if the dimensions of the homology groups over some fields stabilize with n.We prove that this is true when M is odd-dimensional, or when the field is F2 or Q. It is known to be false in the remaining cases.

AB - This paper consists of two related parts. In the first partwegive a self-contained proof of homological stability for the spaces Cn (M; X) of configurations of n unordered points in a connected open manifoldM with labels in a path-connected space X, with the best possible integral stability range of 2* ≤ 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 3, we show that in rational homology the stability range may be improved to * ≤n. In the second part we study to what extent the homology of the spaces Cn (M) can be considered stable when M is a closed manifold. In this case there are no stabilization maps, but one may still ask if the dimensions of the homology groups over some fields stabilize with n.We prove that this is true when M is odd-dimensional, or when the field is F2 or Q. It is known to be false in the remaining cases.

KW - math.AT

KW - 55R40, 55R80

M3 - Journal article

SN - 0033-5606

VL - 64

JO - Quarterly Journal of Mathematics

JF - Quarterly Journal of Mathematics

IS - 1

ER -