A LGEBRAIC MODELS , COHOMOLOGY JUMP LOCI , AND FINITENESS PROPERTIES - PowerPoint PPT Presentation

A LGEBRAIC MODELS , COHOMOLOGY JUMP LOCI , AND FINITENESS PROPERTIES Alex Suciu Northeastern University Topology Seminar University of California, Berkeley October 11, 2017 A LEX S UCIU (N ORTHEASTERN ) M ODELS , JUMP LOCI & FINITENESS UCB T OPOLOGY S EMINAR 1 / 25

F INITENESS PROPERTIES F INITENESS PROPERTIES FOR SPACES AND GROUPS F INITENESS PROPERTIES FOR SPACES AND GROUPS A recurring theme in topology is to determine the geometric and homological finiteness properties of spaces and groups. For instance, to decide whether a path-connected space X is homotopy equivalent to a CW-complex with finite k -skeleton. A group G has property F k if it admits a classifying space K ( G , 1 ) with finite k -skeleton. F 1 : G is finitely generated; F 2 : G is finitely presentable. G has property FP k if the trivial Z G -module Z admits a projective Z G -resolution which is finitely generated in all dimensions up to k . The following implications (none of which can be reversed) hold: G is of type F k ñ G is of type FP k ñ H i ( G , Z ) is finitely generated, for all i ď k ñ b i ( G ) ă 8 , for all i ď k . Moreover, FP k & F 2 ñ F k . A LEX S UCIU (N ORTHEASTERN ) M ODELS , JUMP LOCI & FINITENESS UCB T OPOLOGY S EMINAR 2 / 25

F INITENESS PROPERTIES B IERI –N EUMANN –S TREBEL –R ENZ INVARIANTS B IERI –N EUMANN –S TREBEL –R ENZ INVARIANTS (Bieri–Neumann–Strebel 1987) For a f.g. group G , let Σ 1 ( G ) = t χ P S ( G ) | C χ ( G ) is connected u , where S ( G ) = ( Hom ( G , R ) zt 0 u ) / R + and C χ ( G ) is the induced subgraph of Cay ( G ) on vertex set G χ = t g P G | χ ( g ) ě 0 u . Σ 1 ( G ) is an open set, independent of generating set for G . (Bieri, Renz 1988) � ( Σ k ( G , Z ) = χ P S ( G ) | the monoid G χ is of type FP k . In particular, Σ 1 ( G , Z ) = Σ 1 ( G ) . The Σ -invariants control the finiteness properties of normal subgroups N Ÿ G for which G / N is free abelian: ñ S ( G , N ) Ď Σ k ( G , Z ) N is of type FP k ð where S ( G , N ) = t χ P S ( G ) | χ ( N ) = 0 u . In particular: ñ t˘ χ u Ď Σ 1 ( G ) . ker ( χ : G ։ Z ) is f.g. ð A LEX S UCIU (N ORTHEASTERN ) M ODELS , JUMP LOCI & FINITENESS UCB T OPOLOGY S EMINAR 3 / 25

F INITENESS PROPERTIES B IERI –N EUMANN –S TREBEL –R ENZ INVARIANTS Fix a connected CW-complex X with finite k -skeleton, for some k ě 1. Let G = π 1 ( X , x 0 ) . For each χ P S ( X ) : = S ( G ) , set ! ) λ P Z G | t g P supp λ | χ ( g ) ă c u is finite, @ c P R . y Z G χ = This is a ring, contains Z G as a subring; hence, a Z G -module. (Farber, Geoghegan, Schütz 2010) Σ q ( X , Z ) : = t χ P S ( X ) | H i ( X , y Z G ´ χ ) = 0 , @ i ď q u . ñ Σ q ( G , Z ) = Σ q ( K ( G , 1 ) , Z ) , @ q ď k . (Bieri) G is of type FP k ù A LEX S UCIU (N ORTHEASTERN ) M ODELS , JUMP LOCI & FINITENESS UCB T OPOLOGY S EMINAR 4 / 25

F INITENESS PROPERTIES D WYER –F RIED SETS D WYER –F RIED SETS For a fixed r P N , the connected, regular covers Y Ñ X with group of deck-transformations Z r are parametrized by the Grassmannian of r -planes in H 1 ( X , Q ) . Moving about this variety, and recording when b 1 ( Y ) , . . . , b i ( Y ) are finite defines subsets Ω i r ( X ) Ď Gr r ( H 1 ( X , Q )) , which we call the Dwyer–Fried invariants of X . These sets depend only on the homotopy type of X . Hence, if G is a f.g. group, we may define Ω i r ( G ) : = Ω i r ( K ( G , 1 )) . E XAMPLE Let K be a knot in S 3 . If X = S 3 z K , then dim Q H 1 ( X ab , Q ) ă 8 , and so Ω 1 1 ( X ) = t pt u . But H 1 ( X ab , Z ) need not be a f.g. Z -module. A LEX S UCIU (N ORTHEASTERN ) M ODELS , JUMP LOCI & FINITENESS UCB T OPOLOGY S EMINAR 5 / 25

F INITENESS PROPERTIES D WYER –F RIED SETS T HEOREM Let G be a finitely generated group, and ν : G ։ Z r an epimorphism, with kernel Γ . Suppose Ω k r ( G ) = H , and Γ is of type F k ´ 1 . Then b k ( Γ ) = 8 . Proof: Set X = K ( G , 1 ) ; then X ν = K ( Γ , 1 ) . Since Γ is of type F k ´ 1 , we have b i ( X ν ) ă 8 for i ď k ´ 1. Since Ω k r ( X ) = H , we must have b k ( X ν ) = 8 . It follows that H k ( Γ , Z ) is not f.g., and Γ is not of type FP k . C OROLLARY Let G be a finitely generated group, and suppose Ω 3 1 ( G ) = H . Let ν : G ։ Z be an epimorphism. If the group Γ = ker ( ν ) is finitely presented, then b 3 ( Γ ) = 8 . A LEX S UCIU (N ORTHEASTERN ) M ODELS , JUMP LOCI & FINITENESS UCB T OPOLOGY S EMINAR 6 / 25

F INITENESS PROPERTIES T HE S TALLINGS GROUP T HE S TALLINGS GROUP Let Y = S 1 _ S 1 and X = Y ˆ Y ˆ Y . Clearly, X is a classifying space for G = F 2 ˆ F 2 ˆ F 2 . Let ν : G Ñ Z be the homomorphism taking each standard generator to 1. Set Γ = ker ( ν ) . Stallings (1963) showed that Γ is finitely presented: Γ = x a , b , c , x , y | [ x , a ] , [ y , a ] , [ x , b ] , [ y , b ] , [ a ´ 1 x , c ] , [ a ´ 1 y , c ] , [ b ´ 1 a , c ] y Stallings then showed, via a Mayer-Vietoris argument, that H 3 ( Γ , Z ) is not finitely generated. Alternate explanation: Ω 3 1 ( X ) = H . Thus, by the previous Corollary, a stronger statement holds: b 3 ( Γ ) is not finite. A LEX S UCIU (N ORTHEASTERN ) M ODELS , JUMP LOCI & FINITENESS UCB T OPOLOGY S EMINAR 7 / 25

F INITENESS PROPERTIES B ESTVINA –B RADY GROUPS A RTIN KERNELS AND B ESTVINA -B RADY GROUPS Let G Γ = x v P V ( Γ ) | vw = wv if t v , w u P E ( Γ ) y be the right-angled Artin group associated to a finite simple graph Γ . Given an epimorphism χ : G Γ ։ Z , the corresponding Artin kernel is the group N χ = ker ( χ : G Γ Ñ Z ) . When χ ( v ) = 1 for all v P V ( Γ ) , the group N Γ = N χ is called the Bestvina–Brady group associated to Γ . (Bestvina–Brady 1997) The finiteness properties of N Γ are dictated by the topology of the flag complex ∆ Γ : N Γ is finitely generated ð ñ Γ is connected N Γ is finitely presented ð ñ ∆ Γ is simply-connected. ñ r H i ( ∆ Γ , Z ) = 0 for all i ă k . N Γ is of type FP k ð (Meier–Meinert–VanWyk 1998, Bux–Gonzalez 1999, Papadima–S. 2010) N χ is of type FP k ð ñ dim k H ď k ( N χ , k ) ă 8 , for any field k . A LEX S UCIU (N ORTHEASTERN ) M ODELS , JUMP LOCI & FINITENESS UCB T OPOLOGY S EMINAR 8 / 25

� � � � � F INITENESS PROPERTIES K OLLÁR ’ S QUESTION K OLLÁR ’ S QUESTION Two groups, G 1 and G 2 , are said to be commensurable up to finite kernels if there is a zig-zag of groups and homomorphisms, G 1 H 2 ¨ ¨ ¨ G 2 , H 1 ¨ ¨ ¨ H q with all arrows of finite kernel and cofinite image. Q UESTION (J. K OLLÁR 1995) Given a smooth, projective variety M, is the fundamental group G = π 1 ( M ) commensurable, up to finite kernels, with another group, π , admitting a K ( π , 1 ) which is a quasi-projective variety ? T HEOREM (D IMCA –P APADIMA –S. 2009) For each k ě 3 , there is a smooth, irreducible, complex projective variety M of complex dimension k ´ 1 , such that π 1 ( M ) is of type F k ´ 1 , but not of type FP k . A LEX S UCIU (N ORTHEASTERN ) M ODELS , JUMP LOCI & FINITENESS UCB T OPOLOGY S EMINAR 9 / 25

A LGEBRAIC MODELS AND COHOMOLOGY JUMP LOCI C OMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS C OMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS Let A = ( A ‚ , d ) be a commutative, differential graded algebra over a field k of characteristic 0. A = À i ě 0 A i , where A i are k -vector spaces. The multiplication ¨ : A i b A j Ñ A i + j is graded-commutative, i.e., ab = ( ´ 1 ) | a || b | ba for all homogeneous a and b . The differential d : A i Ñ A i + 1 satisfies the graded Leibnitz rule, i.e., d ( ab ) = d ( a ) b + ( ´ 1 ) | a | a d ( b ) . H ‚ ( A ) inherits an algebra structure from A . A cdga morphism ϕ : A Ñ B is both an algebra map and a cochain map. Hence, it induces a morphism ϕ ˚ : H ‚ ( A ) Ñ H ‚ ( B ) . ϕ is a quasi-isomorphism if ϕ ˚ is an isomorphism. Likewise, ϕ is a q -quasi-isomorphism (for some q ě 1) if ϕ ˚ is an isomorphism in degrees ď q and is injective in degree q + 1. Two cdgas, A and B , are (q-)equivalent ( » q ) if there is a zig-zag of ( q -)quasi-isomorphisms connecting A to B . A LEX S UCIU (N ORTHEASTERN ) M ODELS , JUMP LOCI & FINITENESS UCB T OPOLOGY S EMINAR 10 / 25

A LGEBRAIC MODELS AND COHOMOLOGY JUMP LOCI A LGEBRAIC MODELS FOR SPACES A LGEBRAIC MODELS FOR SPACES Given any (path-connected) space X , there is an associated Sullivan Q -cdga, A PL ( X ) such that H ‚ ( A PL ( X )) = H ‚ ( X , Q ) . An algebraic (q-)model (over k ) for X is a k -cgda ( A , d ) which is ( q -) equivalent to A PL ( X ) b Q k . A cdga A is formal (or just q-formal ) if it is ( q -)weakly equivalent to ( H ‚ ( A ) , d = 0 ) . A CDGA A is of finite-type (or q-finite ) if it is connected (i.e., A 0 = k ¨ 1) and each graded piece A i (with i ď q ) is finite-dimensional. Examples of spaces having finite-type models include: Formal spaces (such as compact Kähler manifolds, hyperplane arrangement complements, toric spaces, etc). Quasi-projective manifolds, compact solvmanifolds, and Sasakian manifolds, etc. A LEX S UCIU (N ORTHEASTERN ) M ODELS , JUMP LOCI & FINITENESS UCB T OPOLOGY S EMINAR 11 / 25