Affiliation:
1. 28264 Max Planck Institute for Mathematics , Vivatsgasse 7 , Bonn , Germany
2. Department of Mathematics , 311551 University of Chicago , 5734 S. University Avenue , Chicago , IL 60637 , USA
Abstract
Abstract
For a compact orientable surface
Σ
g
,
1
\Sigma_{g,1}
of genus 𝑔 with one boundary component and for an odd prime number 𝑝, we study the homology of the unordered configuration spaces
C
∙
(
Σ
g
,
1
)
:
=
∐
n
≥
0
C
n
(
Σ
g
,
1
)
C_{\bullet}(\Sigma_{g,1}):=\coprod_{n\geq 0}C_{n}(\Sigma_{g,1})
with coefficients in
F
p
\mathbb{F}_{p}
.
We describe
H
∗
(
C
∙
(
Σ
g
,
1
)
;
F
p
)
H_{*}(C_{\bullet}(\Sigma_{g,1});\mathbb{F}_{p})
as a bigraded module over the Pontryagin ring
H
∗
(
C
∙
(
D
)
;
F
p
)
H_{*}(C_{\bullet}(D);\mathbb{F}_{p})
, where 𝐷 is a disc, and compute in particular the bigraded dimension over
F
p
\mathbb{F}_{p}
.
We also consider the action of the mapping class group
Γ
g
,
1
\Gamma_{g,1}
and prove that the mod-𝑝 Johnson kernel
K
g
,
1
(
p
)
⊆
Γ
g
,
1
\mathcal{K}_{g,1}(p)\subseteq\Gamma_{g,1}
is the kernel of the action on
H
∗
(
C
∙
(
Σ
g
,
1
;
F
p
)
)
H_{*}(C_{\bullet}(\Sigma_{g,1};\mathbb{F}_{p}))
.
Funder
Horizon 2020 Framework Programme
Danmarks Grundforskningsfond
Engineering and Physical Sciences Research Council