Betti numbers of abelian covers

Betti numbers of abelian covers Alex Suciu Northeastern University Geometry and Topology Seminar University of Wisconsin May 6, 2011 Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 1 / 39

Outline 1 Abelian covers A parameter set for abelian covers Free abelian covers The Dwyer Fried sets 2 Characteristic arrangements and Schubert varieties Characteristic varieties Computing the Ω-invariants Characteristic subspace arrangements Special Schubert varieties 3 Resonance varieties and straight spaces The Aomoto complex Resonance varieties Straight spaces Ω-invariants of straight spaces Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 2 / 39

Outline References S. Papadima and A. Suciu, Bieri Neumann Strebel Renz invariants and homology jumping loci, Proc. London Math. Soc. 100 (2010), no. 3, 795 834. A. Suciu, Resonance varieties and Dwyer Fried invariants, to appear in Advanced Studies Pure Math., Kinokuniya, Tokyo, 2011. A. Suciu, Characteristic varieties and Betti numbers of free abelian covers, preprint 2011. A. Suciu, Y. Yang, and G. Zhao, Homological finiteness of abelian covers, preprint 2011. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 3 / 39

Abelian covers A parameter set for abelian covers Abelian covers Let X be a connected, finite-type CW-complex. We may assume X has a single 0-cell, call it x 0. Set G = π1 (X, x 0 ). H = Gab = H 1 (X, Z). Let A be a finitely generated abelian group. Any epimorphism ν : G A gives rise to a connected, regular cover, X ν X, with group of deck transformations A. Conversely, any conn., regular A-cover p : (Y, y 0 ) (X, x 0 ) yields 1 π 1 (Y, y 0 ) p π 1 (X, x 0 ) ν A 1 so that there is an A-equivariant homeomorphism Y = X ν. Thus, the set of conn., regular A-covers of X can be identified with Epi(G, A)/ Aut(A) = Epi(H, A)/ Aut(A). Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 4 / 39

Abelian covers A parameter set for abelian covers Theorem (S. Yang Zhao 2011) There is a bijection Epi(H, A)/ Aut(A) GL n (Z) P Γ where n = rank H, r = rank A, and P is a parabolic subgroup of GL n (Z). GL n (Z)/ P = Gr n r (Z n ). Γ = Epi(Z n r Tors(H), Tors(A))/ Aut(Tors(A)) is a finite set. GL n (Z) P Γ is the twisted product under the diagonal P-action. This bijection depends on a choice of splitting H = H Tors(H), and a choice of basis for H = H/ Tors(H). Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 5 / 39

Abelian covers Free abelian covers Simplest situation is when A = Z r. Basic example: the universal cover of the r-torus, Z r R r T r. All conn., regular Z r -covers of X arise as pull-backs of this cover: X ν R r X f T r, where f : π 1 (X) π 1 (T r ) realizes the epimorphism ν : G Z r. (Note: X ν is the homotopy fiber of f ). Let ν : Q r H 1 (X, Q) = Q n. We get: { Z r -covers X ν X } { r-planes P ν := im(ν ) in H 1 (X, Q) }. and so Epi(H, Z r )/ Aut(Z r ) = Gr n r (Z n ) = Gr r (Q n ). Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 6 / 39

Abelian covers The Dwyer Fried sets The Dwyer Fried sets Moving about the parameter space for A-covers, and recording how the Betti numbers of those covers vary leads to: Definition The Dwyer Fried invariants of X are the subsets Ω i A (X) = {[ν] Epi(H, A)/ Aut(A) b j(x ν ) <, for j i}. where, X ν X is the cover corresponding to ν : H A. In particular, when A = Z r, Ω i r (X) = { P ν Gr r (H 1 (X, Q)) b j (X ν ) < for j i }, with the convention that Ω i r (X) = if r > n = b 1 (X). For a fixed r > 0, get filtration Gr r (Q n ) = Ω 0 r (X) Ω 1 r (X) Ω 2 r (X). Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 7 / 39

Abelian covers The Dwyer Fried sets The Ω-sets are homotopy-type invariants of X: Lemma Suppose X Y. For each r > 0, there is an isomorphism Gr r (H 1 (Y, Q)) = Gr r (H 1 (X, Q)) sending each subset Ω i r (Y ) bijectively onto Ω i r (X). Thus, we may extend the definition of the Ω-sets from spaces to groups, by setting Ω i r (G) = Ω i r (K (G, 1)) Similarly for the sets Ω i A (X). Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 8 / 39

Abelian covers The Dwyer Fried sets Especially manageable situation: r = n, where n = b 1 (X) > 0. In this case, Gr n (H 1 (X, Q)) = {pt}. This single point corresponds to the maximal free abelian cover, X α X, where α: G G ab = Z n. The sets Ω i n(x) are then given by { {pt} if bj Ω i (X α ) < for j i, n(x) = otherwise. Example Let X = S 1 S k, for some k > 1. Then X α j Z Sk j. Thus, { {pt} for i < k, Ω i n(x) = for i k. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 9 / 39

Abelian covers The Dwyer Fried sets Comparison diagram There is an commutative diagram, Ω i A (X) Epi(G, A)/ Aut A = GL n (Z) P Γ Ω i r (X) Gr r (Q n ) If Ω i r (X) =, then Ω i A (X) =. The above is a pull-back diagram if and only if: If X ν is a Z r -cover with finite Betti numbers up to degree i, then any regular Tors(A)-cover of X ν has the same finiteness property. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 10 / 39

Abelian covers The Dwyer Fried sets Example Let X = S 1 RP 2. Then G = Z Z 2, G ab = Z Z 2, and X α j Z RP 2 j, X ab j Z S 1 j j Z S 2 j. Thus, b 1 (X α ) = 0, yet b 1 (X ab ) =. Hence, Ω 1 1 (X), but Ω1 Z Z 2 (X) =. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 11 / 39

Abelian covers The Dwyer Fried sets Remark Finiteness of the Betti numbers of a free abelian cover X ν does not imply finite-generation of the integral homology groups of X ν. E.g., let K be a knot in S 3, with complement X = S 3 \ K, infinite cyclic cover X ab, and Alexander polynomial K Z[t ±1 ]. Then H 1 (X ab, Z) = Z[t ±1 ]/( K ). Hence, H 1 (X ab, Q) = Q d, where d = deg K. Thus, Ω 1 1 (X) = {pt}. But, if K is not monic, H 1 (X ab, Z) need not be finitely generated. Example (Milnor 1968) Let K be the 5 2 knot, with Alex polynomial K = 2t 2 3t + 2. Then H 1 (X ab, Z) = Z[1/2] Z[1/2] is not f.g., though H 1 (X ab, Q) = Q Q. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 12 / 39

Characteristic arrangements and Schubert varieties Characteristic varieties Characteristic varieties Consider the group of complex-valued characters of G, Ĝ = Hom(G, C ) = H 1 (X, C ) Let G ab = G/G = H 1 (X, Z) be the abelianization of G. The projection ab: G G ab induces an isomorphism Ĝab Ĝ. The identity component, Ĝ0, is isomorphic to a complex algebraic torus of dimension n = rank G ab. The other connected components are all isomorphic to Ĝ 0 = (C ) n, and are indexed by the finite abelian group Tors(G ab ). Ĝ parametrizes rank 1 local systems on X: ρ: G C L ρ the complex vector space C, viewed as a right module over the group ring ZG via a g = ρ(g)a, for g G and a C. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 13 / 39

Characteristic arrangements and Schubert varieties Characteristic varieties The homology groups of X with coefficients in L ρ are defined as H (X, L ρ ) = H (L ρ ZG C ( X, Z)), where C ( X, Z) is the equivariant chain complex of the universal cover of X. Definition The characteristic varieties of X are the sets V i (X) = {ρ Ĝ H j(x, L ρ ) 0, for some j i} Get filtration {1} = V 0 (X) V 1 (X) Ĝ. Each V i (X) is a Zariski closed subset of the algebraic group Ĝ. The characteristic varieties are homotopy-type invariants: Suppose X X. There is then an isomorphism Ĝ = Ĝ, which restricts to isomorphisms V i (X ) = V i (X). Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 14 / 39

Characteristic arrangements and Schubert varieties Characteristic varieties The characteristic varieties may be reinterpreted as the support varieties for the Alexander invariants of X. Let X ab X be the maximal abelian cover. View H (X ab, C) as a module over C[G ab ]. Then (Papadima S. 2010), ( ( V i ( (X) = V ann H j X ab, C ))). Set W i (X) = V i (X) Ĝ0. View H (X α, C) as a module over C[G α ] = Z[t ±1 1,..., t±1 n ], where n = b 1 (G). Then ( ( W i ( (X) = V ann H j X α, C ))). Example Let L = (L 1,..., L n ) be a link in S 3, with complement X = S 3 \ n i=1 L i and Alexander polynomial L = L (t 1,..., t n ). Then j i j i V 1 (X) = {z (C ) n L (z) = 0} {1}. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 15 / 39

Characteristic arrangements and Schubert varieties Computing the Ω-invariants Computing the Ω-invariants Given an epimorphism ν : G Z r, let ˆν : Ẑr Ĝ, ˆν(ρ)(g) = ν(ρ(g)) be the induced monomorphism between character groups. Its image, T ν = ˆν ( Ẑ r ), is a complex algebraic subtorus of Ĝ, isomorphic to (C ) r. Theorem (Dwyer Fried 1987, Papadima S. 2010) Let X be a connected CW-complex with finite k-skeleton, G = π 1 (X). For an epimorphism ν : G Z r, the following are equivalent: 1 The vector space k i=0 H i(x ν, C) is finite-dimensional. 2 The algebraic torus T ν intersects the variety W k (X) in only finitely many points. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 16 / 39

Characteristic arrangements and Schubert varieties Computing the Ω-invariants Let exp: H 1 (X, C) H 1 (X, C ) be the coefficient homomorphism induced by the homomorphism C C, z e z. Lemma Let ν : G Z r be an epimorphism. Under the universal coefficient isomorphism H 1 (X, C ) = Hom(G, C ), the complex r-torus exp(p ν C) corresponds to T ν = ˆν ( Ẑ r ). Thus, we may reinterpret the Ω-invariants, as follows: Theorem Ω i r (X) = { P Gr r (H 1 (X, Q)) dim ( exp(p C) W i (X) ) = 0 }. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 17 / 39

Characteristic arrangements and Schubert varieties Computing the Ω-invariants More generally, Theorem (S. Yang Zhao 2011) Ω i A (X) = { [ν] Epi(H, A)/ Aut(A) im(ˆν) V i (X) is finite }. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 18 / 39

Characteristic arrangements and Schubert varieties Computing the Ω-invariants Corollary Suppose W i (X) is finite. Then Ω i r (X) = Gr r (H 1 (X, Q)), r b 1 (X). Example Let M be a nilmanifold. By (Macinic Papadima 2009): W i (M) = {1}, for all i 0. Hence, Ω i r (M) = Gr r (Q n ), i 0, r n = b 1 (M). Example Let X be the complement of a knot in S m, m 3. Then Ω i 1 (X) = {pt}, i 0. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 19 / 39

Characteristic arrangements and Schubert varieties Computing the Ω-invariants Corollary Let n = b 1 (X). Suppose W i (X) is infinite, for some i > 0. Then Ω q n(x) =, for all q i. Example Let S g be a Riemann surface of genus g > 1. Then Ω i r (S g ) =, for all i, r 1 Ω n r (S g1 S gn ) =, for all r 1 Example Let Y m = m S 1 be a wedge of m circles, m > 1. Then Ω i r (Y m ) =, for all i, r 1 Ω n r (Y m1 Y mn ) =, for all r 1 Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 20 / 39

Characteristic arrangements and Schubert varieties Characteristic subspace arrangements Tangent cones Let W = V (I) be a Zariski closed subset in (C ) n. Definition The tangent cone at 1 to W : TC 1 (W ) = V (in(i)) The exponential tangent cone at 1 to W : τ 1 (W ) = {z C n exp(λz) W, λ C} Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 21 / 39

Characteristic arrangements and Schubert varieties Characteristic subspace arrangements Both types of tangent cones are homogeneous subvarieties of C n ; are non-empty iff 1 W ; depend only on the analytic germ of W at 1; commute with finite unions. Moreover, τ 1 commutes with (arbitrary) intersections; τ 1 (W ) TC 1 (W ) = if all irred components of W are subtori in general (Dimca Papadima S. 2009) τ 1 (W ) is a finite union of rationally defined linear subspaces of C n. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 22 / 39

Characteristic arrangements and Schubert varieties Characteristic subspace arrangements Characteristic subspace arrangements Set n = b 1 (X), and identify H 1 (X, C) = C n and H 1 (X, C ) 0 = (C ) n. Definition The i-th characteristic arrangement of X, denoted C i (X), is the subspace arrangement in H 1 (X, Q) whose complexified union is the exponential tangent cone to W i (X): τ 1 (W i (X)) = L C i (X) L C. We get a sequence C 0 (X), C 1 (X),... of rational subspace arrangements, all lying in H 1 (X, Q) = Q n. The arrangements C i (X) depend only on the homotopy type of X. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 23 / 39

Characteristic arrangements and Schubert varieties Characteristic subspace arrangements Theorem ( Ω i { r (X) P Grr (H 1 (X, Q)) } ) P L {0}. L C i (X) Proof. Fix an r-plane P Gr r (H 1 (X, Q)), and let T = exp(p C). Then: P Ω i r (X) T W i (X) is finite = τ 1 (T W i (X)) = {0} (P C) τ 1 (W i (X)) = {0} P L = {0}, for each L C i (X), Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 24 / 39

Characteristic arrangements and Schubert varieties Characteristic subspace arrangements For straight spaces, the inclusion holds as an equality. If r = 1, the inclusion always holds as an equality. In general, though, the inclusion is strict. E.g., there exist finitely presented groups G for which Ω 1 2 (G) is not open. Example Let G = x 1, x 2, x 3 [x 2 1, x 2], [x 1, x 3 ], x 1 [x 2, x 3 ]x 1 1 [x 2, x 3 ]. Then G ab = Z 3, and V 1 (G) = {1} { t (C ) 3 t 1 = 1 }. Let T = (C ) 2 be an algebraic 2-torus in (C ) 3. Then { T V 1 {1} if T = {t 1 = 1} (G) = C otherwise Thus, Ω 1 2 (G) consists of a single point in Gr 2(H 1 (G, Q)) = QP 2, and so it s not open. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 25 / 39

Characteristic arrangements and Schubert varieties Characteristic subspace arrangements Example Then Let C 1 be a curve of genus 2 with an elliptic involution σ 1. Let C 2 be a curve of genus 3 with a free involution σ 2. Σ 1 = C 1 /σ 1 is a curve of genus 1. Σ 2 = C 2 /σ 2 is a curve of genus 2. We let Z 2 act freely on the product C 1 C 2 via the involution σ 1 σ 2. The quotient space, M = (C 1 C 2 )/Z 2, is a smooth, minimal, complex projective surface of general type with p g (M) = q(m) = 3, and K 2 M = 8. The projection pr 2 : C 1 C 2 C 2 induces a smooth fibration, C 1 M Σ 2. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 26 / 39

Characteristic arrangements and Schubert varieties Characteristic subspace arrangements Example (Continued) Let π = π 1 (M). Then π ab = Z 6, π = (C ) 6, and V 1 (π) = {t t 1 = t 2 = 1} {t 4 = t 5 = t 6 = 1, t 3 = 1}. It follows that Ω 1 2 (π) consists of a single point in Gr 2(Q 6 ), corresponding to the plane spanned by the vectors e 1 and e 2. In particular, Ω 1 2 (π) is not open. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 27 / 39

Characteristic arrangements and Schubert varieties Special Schubert varieties Special Schubert varieties Let V be a homogeneous variety in k n. The set σ r (V ) = { P Gr r (k n ) P V {0} } is a Zariski closed subset of Gr r (k n ), called the variety of incident r-planes to V. When V is a a linear subspace L k n, the variety σ r (L) is called the special Schubert variety defined by L. If L has codimension d in k n, then σ r (L) has codimension d r + 1 in Gr r (k n ). Example The Grassmannian Gr 2 (k 4 ) is the hypersurface in P(k 6 ) with equation p 12 p 34 p 13 p 24 + p 23 p 14 = 0. Let L be a plane in k 4, represented as the row space of a 2 4 matrix. Then σ 2 (L) is the 3-fold in Gr 2 (k 4 ) cut out by the hyperplane p 12 L 34 p 13 L 24 p 23 L 14 + p 14 L 23 p 24 L 13 + p 34 L 12 = 0. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 28 / 39

Characteristic arrangements and Schubert varieties Special Schubert varieties Theorem Ω i ( r (X) Gr r H 1 (X, Q) ) \ ( σ r (L)). L C i (X) Thus, each set Ω i r (X) is contained in the complement of a Zariski closed subset of Gr r (H 1 (X, Q)): the union of the special Schubert varieties corresponding to the subspaces comprising C i (X). Corollary Suppose C i (X) contains a subspace of codimension d. Then Ω i r (X) =, for all r d + 1. Corollary Let X α be the maximal free abelian cover of X. If τ 1 (W 1 (X)) {0}, then b 1 (X α ) =. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 29 / 39

Resonance varieties and straight spaces The Aomoto complex The Aomoto complex Consider the cohomology algebra A = H (X, C), with product operation given by the cup product of cohomology classes. For each a A 1, we have a 2 = 0, by graded-commutativity of the cup product. Definition The Aomoto complex of A (with respect to a A 1 ) is the cochain complex of finite-dimensional, complex vector spaces, (A, a): A 0 a A 1 a A 2 a a A k, with differentials given by left-multiplication by a. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 30 / 39

Resonance varieties and straight spaces The Aomoto complex Alternative interpretation: Pick a basis {e 1,..., e n } for A 1 = H 1 (X, C), and let {x 1,..., x n } be the Kronecker dual basis for A 1 = H 1 (X, C). Identify Sym(A 1 ) with S = C[x 1,..., x n ]. Definition The universal Aomoto complex of A is the cochain complex of free S-modules, : A i C S d i A i+1 C S d i+1 A i+2 C S, where the differentials are defined by d i (u 1) = n j=1 e ju x j for u A i, and then extended by S-linearity. Lemma The evaluation of the universal Aomoto complex at an element a A 1 coincides with the Aomoto complex (A, a). Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 31 / 39

Resonance varieties and straight spaces The Aomoto complex Let X be a connected, finite-type CW-complex. The CW-structure on X is minimal if the number of i-cells of X equals the Betti number b i (X), for every i 0. Equivalently, all boundary maps in C (X, Z) are zero. Theorem (Papadima S. 2010) If X is a minimal CW-complex, the linearization of the cochain complex C (X ab, C) coincides with the universal Aomoto complex of H (X, C). Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 32 / 39

Resonance varieties and straight spaces Resonance varieties Resonance varieties Definition The resonance varieties of X are the sets R i (X) = {a A 1 H j (A, a) 0, for some j i}, defined for all integers 0 i k. Get filtration {0} = R 0 (X) R 1 (X) R k (X) H 1 (X, C) = C n. Each R i (X) is a homogeneous algebraic subvariety of C n. These varieties are homotopy-type invariants of X: If X Y, there is an isomorphism H 1 (Y, C) = H 1 (X, C) which restricts to isomorphisms R i (Y ) = R i (X), for all i 0. (Libgober 2002) TC 1 (W i (X)) R i (X). Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 33 / 39

Resonance varieties and straight spaces Straight spaces Straight spaces Let X be a connected CW-complex with finite k-skeleton. Definition We say X is k-straight if the following conditions hold, for each i k: 1 All positive-dimensional components of W i (X) are algebraic subtori. 2 TC 1 (W i (X)) = R i (X). If X is k-straight for all k 1, we say X is a straight space. The k-straightness property depends only on the homotopy type of a space. Hence, we may declare a group G to be k-straight if there is a K (G, 1) which is k-straight; in particular, G must be of type F k. X is 1-straight if and only if π 1 (X) is 1-straight. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 34 / 39

Resonance varieties and straight spaces Straight spaces Example Let f Z[t] with f (1) = 0. Then X f = (S 1 S 2 ) f e 3 is minimal. W 1 (X f ) = {1}, W 2 (X f ) = V (f ): finite subsets of H 1 (X, C ) = C. R 1 (X f ) = {0}, and R 2 (X f ) = Therefore, X f is always 1-straight, but { {0}, if f (1) 0, C, otherwise. X f is 2-straight f (1) 0. Proposition For each k 2, there is a minimal CW-complex which has the integral homology of S 1 S k and which is (k 1)-straight, but not k-straight. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 35 / 39

Resonance varieties and straight spaces Straight spaces Alternate description of straightness: Proposition The space X is k-straight if and only if the following equalities hold, for all i k: ( ) W i (X) = exp(l C) Z i R i (X) = L C i (X) L C i (X) L C for some finite (algebraic) subsets Z i H 1 (X, C ) 0. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 36 / 39

Resonance varieties and straight spaces Straight spaces Corollary Let X be a k-straight space. Then, for all i k, 1 τ 1 (W i (X)) = TC 1 (W i (X)) = R i (X). 2 R i (X, Q) = L C i (X) L. In particular, the resonance varieties R i (X) are unions of rationally defined subspaces. Example Let G be the group with generators x 1, x 2, x 3, x 4 and relators r 1 = [x 1, x 2 ], r 2 = [x 1, x 4 ][x 2 2, x 3], r 3 = [x 1 1, x 3][x 2, x 4 ]. Then R 1 (G) = {z C 4 z 2 1 2z2 2 = 0}, which splits into two linear subspaces defined over R, but not over Q. Thus, G is not 1-straight. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 37 / 39

Resonance varieties and straight spaces Ω-invariants of straight spaces Ω-invariants of straight spaces Theorem Suppose X is k-straight. Then, for all i k and r 1, Ω i r (X) = Gr r (H 1 (X, Q)) \ σ r (R i (X, Q)). In particular, if all components of R i (X) have the same codimension r, then Ω i r (X) is the complement of the Chow divisor of R i (X, Q). Corollary Let X be k-straight space, with b 1 (X) = n. Then each set Ω i r (X) is the complement of a finite union of special Schubert varieties in Gr r (Q n ). In particular, Ω i r (X) is a Zariski open set in Gr r (Q n ). Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 38 / 39

Resonance varieties and straight spaces Ω-invariants of straight spaces Example Let L = (L 1, L 2 ) be a 2-component link in S 3, with lk(l 1, L 2 ) = 1, and Alexander polynomial L (t 1, t 2 ) = t 1 + t 1 1 1. Let X be the complement of L. Then W 1 (X) (C ) 2 is given by W 1 (X) = {1} {t t 1 = e πi/3 } {t t 1 = e πi/3 } Hence, X is not 1-straight. Since W 1 (X) is infinite, we have Ω 1 2 (X) =. On the other hand, X is non-trivial, and so R 1 (X, Q) = {0}. Hence, σ 2 (R 1 (X, Q)) = {pt}. Alex Suciu (Northeastern University) Betti numbers of abelian covers U. Wisconsin, May 2011 39 / 39