. Chapter, The Long-Moody construction and polynomial functors

. Chapter, Computations of stable homology with twisted coefficients for mapping class groups

, This section recollects Quillen's bracket construction, pre-braided monoidal categories and locally homogeneous categories for the convenience of the reader. It takes up the framework of

, Quillen's bracket construction We fix a strict monoidal groupoid (G, 0)

?. Morphisms, :. , G. Hom, U. , B. )-=-colim et al.,

, A morphism from A to B in the category UG is an equivalence class of pairs (X, f ), where X is an object of G and f : XA ? B is a morphism of G

, ? For all objects X of UG, the identity morphism in the category UG is given by

?. Let and [. X,-f-]-:-a-??-b,

, The strict monoidal category (G, 0) is said to have no zero divisors if for all objects A and B of G, AB ? = 0 if and only if A ? = B ? = 0

, the boundary connected sum along marked half-discs defines a monoidal product on SA, and the 3-disc D 3 is the unit. The braiding of the monoidal structure is given by doing half a Dehn twist in a neighbourhood of the marked half-disc and it is a symmetry

, As for the groupoid ?, the disjoint union of finite sets induces a symmetric monoidal structure (?, 0), the unit 0 being the empty set. We denote by S ? : (N, ?) ? Gr the family of groups

N. , ?. Gr, and S. , surjections {ps n } n?N define a strict monoidal functor PS : SA ? W?. Let PS ?

P. S. ,

H. Byung, K. H. An, and . Ko, A family of representations of braid groups on surfaces, Pacific Journal of Mathematics, vol.247, issue.2, pp.257-282, 2010.

A. Vladimir, The cohomology ring of the colored braid group, Vladimir I. Arnold-Collected Works, pp.183-186, 1969.

A. Vladimir, On some topological invariants of algebraic functions, Vladimir I. Arnold-Collected Works, pp.199-221, 1970.

E. Artin, Theorie der zöpfe, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol.4, pp.47-72, 1925.

J. S. Birman and T. E. Brendle, Braids: a survey. Handbook of knot theory, pp.19-103, 2005.

P. Bellingeri, On presentations of surface braid groups, Journal of Algebra, vol.274, issue.2, pp.543-563, 2004.

J. Bénabou, Introduction to bicategories, Reports of the Midwest Category Seminar, pp.1-77, 1967.

S. Betley, Homology of Gl(R) with coefficients in a functor of finite degree, Journal of Algebra, vol.150, issue.1, pp.73-86, 1992.

S. Betley, Stable K-theory of finite fields. K-theory, vol.17, pp.103-111, 1999.

S. Betley, Twisted homology of symmetric groups, Proceedings of the American Mathematical Society, vol.130, issue.12, pp.3439-3445, 2002.

P. Bellingeri, E. Godelle, and J. Guaschi, Abelian and metabelian quotient groups of surface braid groups, Glasg. Math. J, vol.59, issue.1, pp.119-142, 2017.

S. Bigelow, Braid groups are linear, Journal of the American Mathematical Society, vol.14, issue.2, pp.471-486, 2001.

S. Bigelow, The Lawrence-Krammer representation, Topology and geometry of manifolds, vol.71, pp.51-68, 2001.

S. Bigelow, Homological representations of the Iwahori-Hecke algebra, Proceedings of the Casson Fest, vol.7, pp.493-507, 2004.

J. S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math, vol.22, pp.213-238, 1969.

J. S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, issue.82, 1974.

A. Brownstein and R. Lee, Cohomology of the group of motions of n strings in 3-space, Mapping class groups and moduli spaces of Riemann surfaces, vol.150, pp.51-61, 1991.

K. Søren and . Boldsen, Improved homological stability for the mapping class group with integral or twisted coefficients, Math. Z, vol.270, issue.1-2, pp.297-329, 2012.

E. Brieskorn, Singular elements of semi-simple algebraic groups, Actes du Congres International des Mathématiciens, vol.2, pp.279-284, 1970.

. Kenneth-s-brown, Cohomology of groups, vol.87, 2012.

C. , F. Bödigheimer, and U. Tillmann, Stripping and splitting decorated mapping class groups, Cohomological methods in homotopy theory (Bellaterra, vol.196, pp.47-57, 1998.

S. Bigelow and J. Paul-tian, Generalized Long-Moody representations of braid groups, Commun. Contemp. Math, vol.10, pp.1093-1102, 2008.

C. , F. Bödigheimer, and U. Tillmann, Embeddings of braid groups into mapping class groups and their homology, Configuration spaces, vol.14, pp.173-191, 2012.

W. Burau, Über zopfgruppen und gleichsinnig verdrillte verkettungen, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol.11, pp.179-186, 1935.

F. Callegaro, The homology of the Milnor fiber for classical braid groups, Algebr. Geom. Topol, vol.6, pp.1903-1923, 2006.

T. Church, J. S. Ellenberg, and B. Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J, vol.164, issue.9, pp.1833-1910, 2015.

W. Chen, Homology of braid groups, the Burau representation, and points on superelliptic curves over finite fields, Israel J. Math, vol.220, issue.2, pp.739-762, 2017.

L. Ralph, I. Cohen, and . Madsen, Surfaces in a background space and the homology of mapping class groups, Algebraic geometry-Seattle 2005. Part, vol.1, pp.43-76, 2009.

F. Callegaro, D. Moroni, and M. Salvetti, Cohomology of Artin groups of type?Atype? type?A n , B n and applications, Groups, homotopy and configuration spaces, vol.13, pp.85-104, 2008.

D. E. Cohen, Groups of cohomological dimension one, Lecture Notes in Mathematics, vol.245, 1972.

F. Callegaro and M. Salvetti, Homology of the family of hyperelliptic curves, 2017.

C. Damiani, A journey through loop braid groups, Expositiones Mathematicae, vol.35, issue.3, pp.252-285, 2017.

C. De-concini, C. Procesi, and M. Salvetti, Arithmetic properties of the cohomology of braid groups, Topology, vol.40, issue.4, pp.739-751, 2001.

A. Djament, Sur l'homologie des groupes unitaires à coefficients polynomiaux, Journal of K-theory, vol.10, issue.1, pp.87-139, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00584141

A. Djament, Décomposition de Hodge pour l'homologie stable des groupes d'automorphismes des groupes libres, 2015.

A. Djament, On stable homology of congruence groups, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01565891

A. Djament and C. Vespa, Sur l'homologie des groupes orthogonaux et symplectiques à coefficients tordus, Ann. Sci. Éc. Norm. Supér, vol.43, issue.4, pp.395-459, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00315573

A. Djament and C. Vespa, Sur l'homologie des groupes d'automorphismes des groupes libres à coefficients polynomiaux, Comment. Math. Helv, vol.90, issue.1, pp.33-58, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00741841

A. Djament and C. Vespa, Foncteurs faiblement polynomiaux, To be published in International Mathematics Research Notices, 2017.
URL : https://hal.archives-ouvertes.fr/hal-00851869

S. Eilenberg and S. Lane, On the groups H(?, n). II. Methods of computation, Ann. of Math, vol.60, issue.2, pp.49-139, 1954.

C. J. Earle and A. Schatz, Teichmüller theory for surfaces with boundary, J. Differential Geometry, vol.4, pp.169-185, 1970.

V. Franjou, E. M. Friedlander, A. Scorichenko, and A. Suslin, General linear and functor cohomology over finite fields, Annals of Mathematics-Second Series, vol.150, issue.2, pp.663-728, 1999.

B. Farb and D. Margalit, A Primer on Mapping Class Groups (PMS-49), 2011.

V. Franjou and T. Pirashvili, Stable K-theory is bifunctor homology (after A. Scorichenko)

, Rational representations, the Steenrod algebra and functor homology, vol.16, pp.107-126, 2003.

L. Funar, On the TQFT representations of the mapping class groups, Pacific journal of mathematics, vol.188, issue.2, pp.251-274, 1999.

P. Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France, vol.90, pp.323-448, 1962.

S. Galatius, Stable homology of automorphism groups of free groups, Ann. of Math, vol.173, issue.2, pp.705-768, 2011.

J. Guaschi and . Juan-pineda, A survey of surface braid groups and the lower algebraic K-theory of their group rings, Handbook of Group Actions ii, vol.32, pp.23-76, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00794539

R. Godement, Topologie algébrique et théorie des faisceaux, Actualités Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg, issue.13, 1958.

V. V. Gorjunov, The cohomology of braid groups of series C and D and certain stratifications. Funktsional, Anal. i Prilozhen, vol.12, issue.2, pp.76-77, 1978.

A. Gramain, Le type d'homotopie du groupe des difféomorphismes d'une surface compacte, Annales scientifiques de l'École Normale Supérieure, vol.6, pp.53-66, 1973.

D. Grayson, Higher algebraic K-theory: II (after Daniel Quillen), Lectures Notes in Math, vol.551, pp.217-240, 1976.

G. Gandini and N. Wahl, Homological stability for automorphism groups of RAAGs, Algebr. Geom. Topol, vol.16, issue.4, pp.2421-2441, 2016.

M. Hamstrom, Homotopy groups of the space of homeomorphisms on a 2-manifold. Illinois, J. Math, vol.10, pp.563-573, 1966.

E. Hanbury, Homological stability of non-orientable mapping class groups with marked points, Proceedings of the, vol.137, pp.385-392, 2009.

J. Harer, The third homology group of the moduli space of curves, Duke Math. J, vol.63, issue.1, pp.25-55, 1991.

A. Hatcher, Algebraic topology, vol.606, 2002.

A. Hurwitz, Über riemann'sche flächen mit gegebenen verzweigungspunkten, Mathematische Annalen, vol.39, issue.1, pp.1-60, 1891.

A. Hatcher and N. Wahl, Stabilization for the automorphisms of free groups with boundaries, Geom. Topol, vol.9, pp.1295-1336, 2005.

A. Hatcher and N. Wahl, Stabilization for mapping class groups of 3-manifolds, Duke Math. J, vol.155, issue.2, pp.205-269, 2010.

T. Ishida and M. Sato, A twisted first homology group of the handlebody mapping class group, Osaka J. Math, vol.54, issue.3, pp.587-619, 2017.

T. Ito, The classification of Wada-type representations of braid groups, J. Pure Appl. Algebra, vol.217, issue.9, pp.1754-1763, 2013.

N. V. Ivanov, On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients. In Mapping class groups and moduli spaces of Riemann surfaces, Contemp. Math, vol.150, pp.149-194, 1991.

C. A. Jensen, Homology of holomorphs of free groups, J. Algebra, vol.271, issue.1, pp.281-294, 2004.

C. Jensen, J. Mccammond, and J. Meier, The integral cohomology of the group of loops, Geom. Topol, vol.10, pp.759-784, 2006.

V. Jones, A polynomial invariant for knots via von neumann algebras, Bull. Am. Math. Soc, vol.12, pp.103-111, 1985.

N. Kawazumi, On the stable cohomology algebra of extended mapping class groups for surfaces, Groups of diffeomorphisms, vol.52, pp.383-400, 2008.

T. Kohno, Homological representations of braid groups and kz connections, Journal of Singularities, vol.5, pp.94-108, 2012.

M. Korkmaz, First homology group of mapping class groups of nonorientable surfaces, Math. Proc. Cambridge Philos. Soc, vol.123, issue.3, pp.487-499, 1998.

M. Korkmaz, Low-dimensional homology groups of mapping class groups: a survey, Turkish J. Math, vol.26, issue.1, pp.101-114, 2002.

M. Korkmaz, The symplectic representation of the mapping class group is unique, 2011.

D. Krammer, Braid groups are linear, Ann. of Math, vol.155, issue.2, pp.131-156, 2002.

M. Krannich, Homological stability of topological moduli spaces, 2017.

C. Kassel and V. Turaev, Braid groups, Graduate Texts in Mathematics, vol.247, 2008.

R. J. Lawrence, A topological approach to representations of the Iwahori-Hecke algebra, Internat. J. Modern Phys. A, vol.5, issue.16, pp.3213-3219, 1990.

L. Ruth, Homological representations of the hecke algebra, Communications in mathematical physics, vol.135, issue.1, pp.141-191, 1990.

D. D. Long, On the linear representation of braid groups, Trans. Amer. Math. Soc, vol.311, issue.2, pp.535-560, 1989.

D. D. Long, On the linear representation of braid groups, II. Duke Math. J, vol.59, issue.2, pp.443-460, 1989.

D. D. Long, Constructing representations of braid groups, Comm. Anal. Geom, vol.2, issue.2, pp.217-238, 1994.

I. Marin, On the representation theory of braid groups, Ann. Math. Blaise Pascal, vol.20, issue.2, pp.193-260, 2013.

G. Masbaum, On representations of mapping class groups in integral TQFT, Oberwolfach Reports, vol.5, issue.2, pp.1202-1205, 2008.

S. Lane, Categories for the working mathematician, vol.5, 2013.

I. Madsen and M. Weiss, The stable moduli space of Riemann surfaces: Mumford's conjecture, Ann. of Math, vol.165, issue.2, pp.843-941, 2007.

M. Palmer, A comparison of twisted coefficient systems, 2017.

O. Randal-williams, The homology of the stable nonorientable mapping class group, Algebr. Geom. Topol, vol.8, issue.3, pp.1811-1832, 2008.
DOI : 10.2140/agt.2008.8.1811
URL : http://msp.org/agt/2008/8-3/agt-v8-n3-p24-s.pdf

O. Randal, -. , and N. Wahl, Homological stability for automorphism groups, Adv. Math, vol.318, pp.534-626, 2017.

T. Satoh, Twisted first homology groups of the automorphism group of a free group, Journal of Pure and Applied Algebra, vol.204, issue.2, pp.334-348, 2006.

A. Scorichenko, Stable K-theory and functor homology over a ring, 2000.

A. Soulié, The generalized Long-Moody functors, 2017.

A. Soulié, The Long-Moody construction and polynomial functors, 2017.

A. Soulié, Computations of homology with twisted coefficients for mapping class groups, 2018.

M. Stukow, The twist subgroup of the mapping class group of a nonorientable surface, Osaka Journal of Mathematics, vol.46, issue.3, pp.717-738, 2009.

M. Stukow, The first homology group of the mapping class group of a nonorientable surface with twisted coefficients, Topology Appl, vol.178, pp.417-437, 2014.

. Richard-g-swan, Groups of cohomological dimension one, Journal of Algebra, vol.12, issue.4, pp.585-610, 1969.

D. Tong, S. Yang, and Z. Ma, A new class of representations of braid groups, Communications in Theoretical Physics, vol.26, issue.4, pp.483-486, 1996.

V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, Translations of Mathematical Monographs, vol.98, 1992.

. Wilberd-van-der-kallen, Homology stability for linear groups. Inventiones mathematicae, vol.60, pp.269-295, 1980.

V. V. Vershinin, Homology of braid groups and their generalizations, Knot theory, vol.42, pp.421-446, 1995.
DOI : 10.4064/-42-1-421-446
URL : https://www.impan.pl/shop/publication/transaction/download/product/110294?download.pdf

C. Vespa, La catégorie F quad des foncteurs de Mackey généralisés pour les formes quadratiques sur F 2, 2005.
DOI : 10.1016/j.crma.2006.06.032

C. Vespa, La catégorie de foncteurs F quad sur les espaces quadratiques sur F 2, Comptes Rendus Mathematique, vol.343, issue.4, pp.229-234, 2006.
DOI : 10.1016/j.crma.2006.06.032

C. Vespa, Generic representations of orthogonal groups: the functor category F quad, Journal of Pure and Applied Algebra, vol.212, issue.6, pp.1472-1499, 2008.
DOI : 10.4064/fm200-3-2
URL : https://www.impan.pl/shop/publication/transaction/download/product/89304?download.pdf

, Out(F n ) and everything in between: automorphism groups of RAAGs, Groups St Andrews, vol.422, pp.105-127, 2013.

M. Wada, Group invariants of links, Topology, vol.31, issue.2, pp.399-406, 1992.

B. Wajnryb, Artin groups and geometric monodromy, Inventiones mathematicae, vol.138, issue.3, pp.563-571, 1999.

C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol.38, 1994.

C. H. Jennifer and . Wilson, Representation stability for the cohomology of the pure string motion groups, Algebr. Geom. Topol, vol.12, issue.2, pp.909-931, 2012.