Configuration spaces on a wedge of spheres and Hochschild–Pirashvili homology

We study the compactly supported rational cohomology of configuration spaces of points on wedges of spheres, equipped with natural actions of the symmetric group and the group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">Out</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of outer automorphisms of the free group. These representations show up in seemingly unrelated parts of mathematics, from cohomology of moduli spaces of curves to polynomial functors on free groups and Hochschild–Pirashvili cohomology. We show that these cohomology representations form a polynomial functor, and use various geometric models to compute many of its composition factors. We further compute the composition factors completely for all configurations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>10</mml:mn> </mml:mrow> </mml:math> points. An application of this analysis is a new super-exponential lower bound on the symmetric group action on the weight <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>0</mml:mn> </mml:math> component of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>c</mml:mi> <mml:mo>*</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ℳ</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> .