HOMOLOGICAL STABILITY, REPRESENTATION STABILITY, AND FI-MODULES

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1 HOMOLOGICAL STABILITY, REPRESENTATION STABILITY, AND FI-MODULES THOMAS CHURCH ABSTRACT. Homological stability is the classical phenomenon that for many natural families of moduli spaces the homology groups stabilize. Often the limit is the homology of another interesting space; for example, the homology of the braid groups converges to the homology of the space of self-maps of the Riemann sphere. Representation stability makes it possible to extend this to situations where classical homological stability simply does not hold, using ideas inspired by asymptotic representation theory. I will give a broad survey of homological stability and a gentle introduction to the tools and results of representation stability, focusing on its applications in topology. Part I: homological stability Part II: representation stability Y n Y n+1 H (Y n ) H (Y n ) is isomorphism for f (n). H (SO(n)) Date: 26 August

2 2 THOMAS CHURCH SO(1) SO(2) SO(3) SO(4) SO(5) SO(6) SO( ) H 0 Z Z Z Z Z Z Z H 1 Z Z/2 Z/2 Z/2 Z/2 Z/2 H H 3 Z Z 2 Z Z/2 Z Z/2 Z Z/2 H 4 Z/2 Z/2 Z/2 Z/2 H 5 0 Z/2 Z Z/2 Z/2 Z/2 H 6 Z Z/2 Z/2 Z/2 Z/2 Z/2 H 7 Z Z Z Z/2 Z/2 H 8 Z/2 Z Z/2 Z/2... H 9 0 Z/2... H 10 Z Z/2 Z/2... SO(n) SO(n + 1) SO(n) SO(n + 1) is (n 1)-connected. S n H (SO(n)) H (SO(n + 1)) for < n 1. Often there s some interesting Y such that H (Y n ) = H (Y), n (for example, SO( )). Configuration space Conf n (M) = {S M : S = n}. Example. Conf n (C) is a K(Braid n, 1). H (Conf n (C)) = H (Braid n ) Theorem (Arnold 1969). H (Conf n (C)) H (Conf n+1 (C)) is an isomorphism for n 2. Audience question: what is the map? Add a point very far away from the other points. Theorem (F. Cohen 1973). lim n H (Conf n (C)) = H (R 2, R 2 ). Warning: π 1 (R 2, R 2 ) = Z = π 1 Conf C = Braid.

3 HOMOLOGICAL STABILITY, REPRESENTATION STABILITY, AND FI-MODULES 3 Stability for GL n Z (originally theorem of Charney) New approach Bestvina Church, inspired by Hatcher Vogtmann. Define constants c n as follows: P n = simplicial complex with vertices pairs (a, b) in Z n with a 1 = 1, simplices on (a 1, b 1 ),..., (a k, b k ) if a i b j = 1 if i = j and 0 otherwise. c n = connectivity of P n P n is c n -connected. Conjecture (Church Bestvina). P n S n 2, so c n = n 3. X n = { inner products ω on R n } = { positive definite symmetric n n matrices R (n+1 2 ) } nonpositively curved metric GL n R X n GL n Z X n with compact = finite stabilizers. H (GL n Z; Q) = H (X n / GL n Z; Q), Y n = X n / GL n Z Given ω, say v Z n is ω-integral if ω(v, v) = 1 and ω(v, Z n ) Z. Define X k n = {ω X n # of ω-integral vectors > n k}. X 1 n X 2 n X n n X n+1 n = X n. The filtration starts with lots of ω-integral vectors, ends with one ω-integral vector. Theorem (C. Bestvina). (1) Xn k is c k -connected, so H (GL n Z; Q) = H (Xn/ k GL n Z; Q) for c k. (2) The quotient space Xn/ k GL n Z is independent of n for n k 1. (If conjecture of BC = H (GL n Z; Q) independent of n for n > + 1.)

4 4 THOMAS CHURCH (3) Integrally H (GL n Z) = H orb (Xn/ k GL n Z). As orbifold X k n/ GL n Z not constant by stabilizers for G k O n k (Z). starting again from scratch... Often guess a space Y and prove lim n H (Y n ) = H (Y) without knowing that H (Y n ) actually stabilize. Recall: if G is a discrete group topology: when points collide, you multiply the labels, and when points hit they boundary, they disappear. BG is a K(G, 1), π 1 = G, π i = 0. What if M monoid, like M = N (all you need in the definition of BG is multiplication). What is BN? π 1 BN = N because π 1 is a group. It turns out that π 1 BN is Z, in fact BN = K(Z, 1). BM = K(M, 1), where M is groupified M (questioned by audience, need sufficiently nice M). In general, ΩBG is a K(G, 0), which is to say ΩBG G. For M discrete, ΩBM = M. In general, ΩBM is groupification of M

5 HOMOLOGICAL STABILITY, REPRESENTATION STABILITY, AND FI-MODULES 5 M FinSubsets(R 2 ) FinSubsets(R ) Surf (R ) = {U R }{U smooth connected 2- manifold w/ 1 component boundary Metric graphs = { isometry classes of metric graphs w/ basepoint }? Subspaces(C ) ΩBM (R 2, R 2 ) (F.Cohen) (R, R ) (Barratt Priddy Quillen) (R, Aff 2 (R )) (Madsen Weiss) (R, Graphs(R )) (Galatius) ( = (R, R )) (R 1, GL C) (Bott periodicity) The entries in left column are equivalent to n N Conf n R 2, n N BS n, g N BDiff(Σ g ), n N BAut(F n ), n N BU(n), respectively. If M = M n, then Z M ΩBM is H -iso. Theorem (Arnold). H (Conf n (C), Z) H (Conf n+1 (C); Z) stabilizes. Theorem (McDuff, Segal). For any open manifold M, stabilizes for n. H (Conf n (M); Z) H (Conf n+1 (M); Z) Theorem (Church 2012). For any M, stabilizes for n >. H (Conf n (M); Q) H (Conf n+1 (M); Q) Why the historical gap? (1) No maps Conf n (M) Conf n+1 (M). (2) False integrally (Artin 1925): H 1 (Conf n (S 2 ); Z) = Z/(2n 2)Z S n PConf n (M) = M n {p i = p j }, and PConf n (M) Conf n (M). H (PConf n (M); Q) S n H (Conf n (M); Q) reduces to understanding H of PConf n (M) as an S n -representation.

6 6 THOMAS CHURCH Now we have maps PConf n (M) PConf n+1 (M) (forget the last point). H (SL n Z; Z) stabilizes, SL n (Z, l) = ker(sl n Z SL n Z/l) = {M id mod l} but H doesn t stabilize, H 1 SL(Z, Z/l) = sl n Z/l = (Z/l) n2 1. Define T [n] SL T Z = {M SL n Z M ij = δ ij if i or j / T}. SL T (Z, l) = {M SL n (Z, l) M ij = δ ij if i or j / T}. S T, SL S Z SL t Z. Theorem (Church Ellenberg, after Putman CEFN). inductive stability n, H i (SL n (Z, l); Z) = lim T [n], T 2 i+2 H i(sl T (Z, l); Z) H i (Γ n ) = lim T C,T [n] H i (Γ T ) Definition. FI = category of Finite Sets T and Injections S T. FI-module, V : FI Z-Mod. Example. SL Z : FI Groups SL (Z, l) : FI Groups H i (SL (Z, l)) : FI Z-Mod is an FI-module. PConf M : FI Spaces op, T PConf T (M) = Inj(T, M). H i PConf M : FI Z-Mod is an FI-module. An FI-module V is for each n abelian group V n with S n -action, S n V n, with maps V n V N (n < N) that play nicely with S n -action. As algebraic objects, notion of finitely generated FI-module, similarly finitely presented, projective, injective etc.

7 HOMOLOGICAL STABILITY, REPRESENTATION STABILITY, AND FI-MODULES 7 Theorem (CEF). If V an FI-module over Q, then the following are equivalent: (1) representation stability for S n -representations V n (2) V is finitely generated. Theorem (CE). Any V, the following are equivalent: (1) inductive stability : C such that V n = lim T C V T (2) V is finitely presented Theorem (CEFN). FI is Noetherian, so finitely generated = finitely presented, and both are preserved by e.g. spectral sequences. Theorem (Church Putman). k, C k, genus g such that k-th term of Johnson filtration is generated by elements supported on surfaces / splittings of genus C k. Is SL Z finitely presented as FI-group? Does there exist C such that SL n Z = lim T C SL T Z?