Skip to main content

Steve Halperin

Follower

1

Following

1

Public Views

Alexander A Voronov

University of Minnesota - Twin Cities

University of Neuchâtel

Ohio State University

Hvedri Inassaridze

Mamuka Jibladze

Teimuraz Pirashvili

Emzar Khmaladze

antonio Gómez-Tato

University of Santiago de Compostela

Interests

Uploads

Papers by Steve Halperin

Lectures on minimal models

Mémoires de la Société mathématique de France, 1983

CkapteA 4 ChapteA 5 Notation and conventions. KS^extensions : Definitions, space of indecomposabl... more CkapteA 4 ChapteA 5 Notation and conventions. KS^extensions : Definitions, space of indecomposables, contractible KS complexes, pullbacks. ReducjCion_ to a_minimal extension. The structure of a minimal extension. Morphisms of extensions. Homotopies and liftings : The c.g.d.a. ((C,B) ,D), the inclusions C-»• (C,B) , homotopy and based ho-\ .X, o 1 motopy (rel B), lifting of morphisms the extension (C,B) 31-(C,B) 1-AQ (thms 5.19. 5.20).

Aspherical completions and rationally inert elements

arXiv (Cornell University), Apr 18, 2019

Let X be a connected space. An element [f ] ∈ π n (X) is called rationally inert if π * (X) ⊗ Q →... more Let X be a connected space. An element [f ] ∈ π n (X) is called rationally inert if π * (X) ⊗ Q → π * (X ∪ f D n+1) ⊗ Q is surjective. We extend the results of [16] and prove in particular that if X ∪ f D n+1 is a Poincaré duality complex and the algebra H(X) requires at least two generators then [f ] ∈ π n (X) is rationally inert. On the other hand, if X is rationally a wedge of at least two spheres and f is rationally non trivial, then f is rationally inert. Finally if f is rationally inert then the rational homotopy of the homotopy fibre of the injection X → X ∪ f D n+1 is the completion of a free Lie algebra.

Enriched Differential Lie Algebras in Topology

arXiv (Cornell University), Aug 25, 2022

The context of this article is the Sullivan rationalization X → X Q of path connected topological... more The context of this article is the Sullivan rationalization X → X Q of path connected topological spaces. This is constructed by Sullivan via his Sullivan algebras, which are commutative differential graded algebras (cdga's), (∧V, d), in which: (i) V = V ≥1 , (ii) ∧V is the free graded commutative algebra generated by V , and (iii) d satisfies a certain "Sullivan condition". As Sullivan realized from the outset, the component, d 1 : V → V ∧ V of d, defines by duality a Lie bracket in L V := s −1 Hom(V ; Q); (s −1 is inverse suspension). Moreover, L V is a graded Lie algebra, the homotopy Lie algebra of (∧V, d). We recall now that each connected space admits a quasi-isomorphism (∧V, d)

The ranks of the homotopy groups of a finite complex

Sullivan completions

arXiv: Algebraic Topology, 2019

The Sullivan construction associates to each path connected space or connected simplicial set, $X... more The Sullivan construction associates to each path connected space or connected simplicial set, $X$, a special cdga, its minimal model $(\land V,d)$, and to each such cdga $\land W$ its geometric realisation $\langle \land W\rangle$. The composite of these constructions is the Sullivan completion, $X_{\mathbb Q}$, of $X$. In this paper we give a survey of the main properties of Sullivan completions, and include explicit examples.

Lie Algebras of Polynomial Growth

Journal of the London Mathematical Society, 1991

Kac has introduced the notion of (polynomial) growth for a graded Lie algebra. Here we consider L... more Kac has introduced the notion of (polynomial) growth for a graded Lie algebra. Here we consider Lie algebras L that occur as ideals either in the rational homotopy Lie algebra of a simply connected CW complex of finite type and finite category or as ideals in the homotopy Lie algebra of a local noetherian ring. Theorem. If these ideals have {finite) polynomial growth, then they are finite dimensional.

Engel elements in the homotopy Lie algebra

Journal of Algebra, 1991

The linear span of these elements is a graded subspace Ec L. The depth of L is the intinum (possi... more The linear span of these elements is a graded subspace Ec L. The depth of L is the intinum (possibly co) of the set of integers m for which ExtEL(k, UL) #O. It is known that if L is the rational homotopy Lie algebra of a simply connected space, X, or the homotopy Lie algebra of a local noetherian ring, A, then the depth of L is bounded above respectively by the Lusternik-Schnirelmann category of X and the embedding dimension of A. THEOREM. If L is concentrated in degrees >0 (or in degrees <0) and if depth L = m then there are at most m linearly independent Engel elements of even degree: 1 dim, ESi < depth L.

Hopf Algebras and a Counterexample to a Conjecture of Anick

Journal of Algebra, 1994

The category of a map and the grade of a module

Israel Journal of Mathematics, 1992

Letf: Y → X be a continuous map between connectedCW complexes. The homologyH * (F) of the homotop... more Letf: Y → X be a continuous map between connectedCW complexes. The homologyH * (F) of the homotopy fibre is then a module over the loop space homologyH * (ΩX). Theorem:If H * (F; R) and H * (ΩX; R) are R-free (R a principal ideal domain) then for some H * (ΩX; R)-projective module P=P&amp;gt;0 and for some m

The homotopy lie algebra for finite complexes

Publications mathématiques de l'IHÉS, 1982

L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.f... more L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ THE HOMOTOPY LIE ALGEBRA FOR FINITE COMPLEXES by YVES FfiLIX (1), STEPHEN HALPERIN (2) and JEAN-CLAUDE THOMAS (3) i. Introduction.-A generic question in topology asks how geometric restrictions on a topological space S are reflected in restrictions on^(S). A classical example is this: which discrete groups G admit a finite GW complex as classifying space? In this paper we shall deal with an analogous question for Lie algebras and simply connected spaces. Henceforth, and throughout this paper, we shall consider only those spaces which are simply connected and have the homotopy type ofGW complexes whose rational homology is finite dimensional in each degree. Such spaces will be called i-connected GW spaces of finite Qj-tyRe. For such spaces 7Tp(S)®Q^ is finite dimensional (each p), and the Whitehead product in ^(S), transferred to T^(OS) by the canonical isomorphism, makes T^(QS) ® Q, into a connected graded Lie algebra of finite type {i.e. finite dimensional in each degree): the rational homotopy Lie algebra of S. A striking result of Quillen [QJ asserts that every connected graded Lie algebra (over QJ of finite type arises in this way. The situation for finite complexes is very different, and the question referred to above, which forms the starting point of this paper, can be stated as the Problem.-What restrictions are imposed on the rational homotopy Lie algebra of a space S, if S is a finite, i-connected, GW complex? We shall establish serious restrictions, both on the integers dimTTp(S) ®Q,, and on the Lie structure. These restrictions, moreover, turn out to hold for the much larger class of those i-connected GW spaces of finite Q^-type whose rational Lusternik-Schnirelmann category is finite. Recall that the Lusternik Schnirelmann category of a space S, as normalized by Ganea [Ga], is the least integer m such that S can be covered by m + i open sets, each (1) Chercheur qualifie au F.N.R.S. (2) During this research the second named author enjoyed the hospitality of the Sonderforschungsbereich (40) Mathematik at the University of Bonn,

Growth and Lie brackets in the homotopy Lie algebra

Homology, Homotopy and Applications, 2002

Let L be an infinite dimensional graded Lie algebra that is either the homotopy Lie algebra π * (... more Let L be an infinite dimensional graded Lie algebra that is either the homotopy Lie algebra π * (ΩX) ⊗ Q for a finite ndimensional CW complex X, or else the homotopy Lie algebra for a local noetherian commutative ring R (U L = Ext R (Ik, Ik)) in which case put n = (embdim − depth)(R). Theorem: (i) The integers λ k = k+n−2 q=k dimL i grow faster than any polynomial in k. (ii) For some finite sequence x 1 ,. .. , x d of elements in L and some N , any y ∈ L N satisfies: some [x i , y] = 0.

Elliptic spaces

Bulletin of the American Mathematical Society, 1991

A simply connected CW complex of finite type is elliptic if it has finite L.S.-category and if it... more

Graded Lie algebras with finite polydepth

Annales Scientifiques de l’École Normale Supérieure, 2003

If A is a graded connected algebra then we define a new invariant, polydepth A, which is finite i... more If A is a graded connected algebra then we define a new invariant, polydepth A, which is finite if Ext * A (M, A) = 0 for some A-module M of at most polynomial growth. THEOREM 1: If f : X → Y is a continuous map of finite category, and if the orbits of H * (ΩY) acting in the homology of the homotopy fibre grow at most polynomially, then H * (ΩY) has finite polydepth. THEOREM 5: If L is a graded Lie algebra and polydepth UL is finite then either L is solvable and UL grows at most polynomially or else for some integer d and all r, k+d i=k+1 dim Li k r , k some k(r).

Rational Homotopy Theory via Sullivan Models: A Survey

Notices of the International Congress of Chinese Mathematicians

the last 40 years, and most especially to our good friend, Jean-Claude Thomas, frequently our co-... more the last 40 years, and most especially to our good friend, Jean-Claude Thomas, frequently our co-author, including in particular for the two volumes on rational homotopy. Contents 1 Notation and terminology 2 Sullivan models 3 The spatial realization of a Sullivan algebra 4 Homotopy groups 5 The homotopy Lie algebra and the fundamental group 6 Whitehead products and nilpotence 7 Hurewicz homomorphisms 8 Rationalizations, Malcev completions and Sullivan spaces 9 LS category 10 The trichotomy : growth of homotopy groups 11 Minimal ∧-models 12 Minimal ∧-models of a fibration 13 Holonomy representations 14 Fiber squares 15 Mapping spaces 16 Acyclic closures and loop spaces 17 Depth 18 Configuration spaces 19 The complement of arrangements 20 The group of homotopy self-equivalences of a space

Enriched Lie algebras in topology, I

To each path connected space X the Sullivan theory of minimal models associates a commutative dif... more To each path connected space X the Sullivan theory of minimal models associates a commutative differential graded algebra, its minimal model (∧V, d), and with it a graded Lie algebra LX that is the homotopy Lie algebra of its geometric realization < ∧V, d >. When X is a simply connected space with finite Betti numbers, then LX is isomorphic to the Lie algebra of X, π∗(ΩX)⊗Q, equipped with the Whitehead bracket. In the general case, LX is a complete enriched Lie algebra: An enriched Lie algebra is a graded Lie algebra L = L≥0 together with a family of ideals Iα indexed by a partially ordered set with α ≥ β if and only if Iα ⊂ Iβ, and such that each quotient L/Iα is a finite dimensional nilpotent Lie algebra. The enriched Lie algebra (L, {Iα}) is complete if L = lim ←−α L/Iα. Enriched Lie algebras are the natural extension of graded Lie algebras for path connected spaces, because each complete enriched Lie algebra L is the homotopy Lie algebra of a path connected space X. This t...

A note on Gorenstein spaces

Associated with an augmented differential graded algebra R= R^≥ 0 is a homotopy invariant T(R). T... more Associated with an augmented differential graded algebra R= R^≥ 0 is a homotopy invariant T(R). This is a graded vector space, and if H^0(R) is the ground field and H^>N(R)= 0 then dim T(R)= 1 if and only if H(R) is a Poincaré duality algebra. In the case of Sullivan extensions W→ W⊗ Z→ Z in which dim H( Z)<∞ we show that T( W⊗ Z)= T( W)⊗ T( Z). This is applied to finite dimensional CW complexes X where the fundamental group G acts nilpotently in the cohomology H(X; Q) of the universal covering space. If H(X; Q) is a Poincaré duality algebra and H(X; Q) and H(BG; Q) are finite dimensional then they are also Poincaré duality algebras.

Homotopy Lie algebras and Sullivan Lie algebras

Depth of a connected graded Lie algebra of finite type

Structure of a graded Lie algebra of finite depth

The depth of a Riemann surface and of a right-angled Artin group

Journal of Homotopy and Related Structures, 2019

We consider two families of spaces, X : the closed orientable Riemann surfaces of genus g > 0 and... more We consider two families of spaces, X : the closed orientable Riemann surfaces of genus g > 0 and the classifying spaces of right-angled Artin groups. In both cases we compare the depth of the fundamental group with the depth of an associated Lie algebra, L, that can be determined by the minimal Sullivan algebra. For these spaces we prove that depth Q[π 1 (X)] = depth L and give precise formulas for the depth.

Lectures on minimal models

Mémoires de la Société mathématique de France, 1983

CkapteA 4 ChapteA 5 Notation and conventions. KS^extensions : Definitions, space of indecomposabl... more CkapteA 4 ChapteA 5 Notation and conventions. KS^extensions : Definitions, space of indecomposables, contractible KS complexes, pullbacks. ReducjCion_ to a_minimal extension. The structure of a minimal extension. Morphisms of extensions. Homotopies and liftings : The c.g.d.a. ((C,B) ,D), the inclusions C-»• (C,B) , homotopy and based ho-\ .X, o 1 motopy (rel B), lifting of morphisms the extension (C,B) 31-(C,B) 1-AQ (thms 5.19. 5.20).

Aspherical completions and rationally inert elements

arXiv (Cornell University), Apr 18, 2019

Let X be a connected space. An element [f ] ∈ π n (X) is called rationally inert if π * (X) ⊗ Q →... more Let X be a connected space. An element [f ] ∈ π n (X) is called rationally inert if π * (X) ⊗ Q → π * (X ∪ f D n+1) ⊗ Q is surjective. We extend the results of [16] and prove in particular that if X ∪ f D n+1 is a Poincaré duality complex and the algebra H(X) requires at least two generators then [f ] ∈ π n (X) is rationally inert. On the other hand, if X is rationally a wedge of at least two spheres and f is rationally non trivial, then f is rationally inert. Finally if f is rationally inert then the rational homotopy of the homotopy fibre of the injection X → X ∪ f D n+1 is the completion of a free Lie algebra.

Enriched Differential Lie Algebras in Topology

arXiv (Cornell University), Aug 25, 2022

The context of this article is the Sullivan rationalization X → X Q of path connected topological... more The context of this article is the Sullivan rationalization X → X Q of path connected topological spaces. This is constructed by Sullivan via his Sullivan algebras, which are commutative differential graded algebras (cdga's), (∧V, d), in which: (i) V = V ≥1 , (ii) ∧V is the free graded commutative algebra generated by V , and (iii) d satisfies a certain "Sullivan condition". As Sullivan realized from the outset, the component, d 1 : V → V ∧ V of d, defines by duality a Lie bracket in L V := s −1 Hom(V ; Q); (s −1 is inverse suspension). Moreover, L V is a graded Lie algebra, the homotopy Lie algebra of (∧V, d). We recall now that each connected space admits a quasi-isomorphism (∧V, d)

The ranks of the homotopy groups of a finite complex

Sullivan completions

arXiv: Algebraic Topology, 2019

The Sullivan construction associates to each path connected space or connected simplicial set, $X... more The Sullivan construction associates to each path connected space or connected simplicial set, $X$, a special cdga, its minimal model $(\land V,d)$, and to each such cdga $\land W$ its geometric realisation $\langle \land W\rangle$. The composite of these constructions is the Sullivan completion, $X_{\mathbb Q}$, of $X$. In this paper we give a survey of the main properties of Sullivan completions, and include explicit examples.

Lie Algebras of Polynomial Growth

Journal of the London Mathematical Society, 1991

Kac has introduced the notion of (polynomial) growth for a graded Lie algebra. Here we consider L... more Kac has introduced the notion of (polynomial) growth for a graded Lie algebra. Here we consider Lie algebras L that occur as ideals either in the rational homotopy Lie algebra of a simply connected CW complex of finite type and finite category or as ideals in the homotopy Lie algebra of a local noetherian ring. Theorem. If these ideals have {finite) polynomial growth, then they are finite dimensional.

Engel elements in the homotopy Lie algebra

Journal of Algebra, 1991

The linear span of these elements is a graded subspace Ec L. The depth of L is the intinum (possi... more The linear span of these elements is a graded subspace Ec L. The depth of L is the intinum (possibly co) of the set of integers m for which ExtEL(k, UL) #O. It is known that if L is the rational homotopy Lie algebra of a simply connected space, X, or the homotopy Lie algebra of a local noetherian ring, A, then the depth of L is bounded above respectively by the Lusternik-Schnirelmann category of X and the embedding dimension of A. THEOREM. If L is concentrated in degrees >0 (or in degrees <0) and if depth L = m then there are at most m linearly independent Engel elements of even degree: 1 dim, ESi < depth L.

Hopf Algebras and a Counterexample to a Conjecture of Anick

Journal of Algebra, 1994

The category of a map and the grade of a module

Israel Journal of Mathematics, 1992

Letf: Y → X be a continuous map between connectedCW complexes. The homologyH * (F) of the homotop... more Letf: Y → X be a continuous map between connectedCW complexes. The homologyH * (F) of the homotopy fibre is then a module over the loop space homologyH * (ΩX). Theorem:If H * (F; R) and H * (ΩX; R) are R-free (R a principal ideal domain) then for some H * (ΩX; R)-projective module P=P&amp;gt;0 and for some m

The homotopy lie algebra for finite complexes

Publications mathématiques de l'IHÉS, 1982

L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.f... more L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ THE HOMOTOPY LIE ALGEBRA FOR FINITE COMPLEXES by YVES FfiLIX (1), STEPHEN HALPERIN (2) and JEAN-CLAUDE THOMAS (3) i. Introduction.-A generic question in topology asks how geometric restrictions on a topological space S are reflected in restrictions on^(S). A classical example is this: which discrete groups G admit a finite GW complex as classifying space? In this paper we shall deal with an analogous question for Lie algebras and simply connected spaces. Henceforth, and throughout this paper, we shall consider only those spaces which are simply connected and have the homotopy type ofGW complexes whose rational homology is finite dimensional in each degree. Such spaces will be called i-connected GW spaces of finite Qj-tyRe. For such spaces 7Tp(S)®Q^ is finite dimensional (each p), and the Whitehead product in ^(S), transferred to T^(OS) by the canonical isomorphism, makes T^(QS) ® Q, into a connected graded Lie algebra of finite type {i.e. finite dimensional in each degree): the rational homotopy Lie algebra of S. A striking result of Quillen [QJ asserts that every connected graded Lie algebra (over QJ of finite type arises in this way. The situation for finite complexes is very different, and the question referred to above, which forms the starting point of this paper, can be stated as the Problem.-What restrictions are imposed on the rational homotopy Lie algebra of a space S, if S is a finite, i-connected, GW complex? We shall establish serious restrictions, both on the integers dimTTp(S) ®Q,, and on the Lie structure. These restrictions, moreover, turn out to hold for the much larger class of those i-connected GW spaces of finite Q^-type whose rational Lusternik-Schnirelmann category is finite. Recall that the Lusternik Schnirelmann category of a space S, as normalized by Ganea [Ga], is the least integer m such that S can be covered by m + i open sets, each (1) Chercheur qualifie au F.N.R.S. (2) During this research the second named author enjoyed the hospitality of the Sonderforschungsbereich (40) Mathematik at the University of Bonn,

Growth and Lie brackets in the homotopy Lie algebra

Homology, Homotopy and Applications, 2002

Let L be an infinite dimensional graded Lie algebra that is either the homotopy Lie algebra π * (... more Let L be an infinite dimensional graded Lie algebra that is either the homotopy Lie algebra π * (ΩX) ⊗ Q for a finite ndimensional CW complex X, or else the homotopy Lie algebra for a local noetherian commutative ring R (U L = Ext R (Ik, Ik)) in which case put n = (embdim − depth)(R). Theorem: (i) The integers λ k = k+n−2 q=k dimL i grow faster than any polynomial in k. (ii) For some finite sequence x 1 ,. .. , x d of elements in L and some N , any y ∈ L N satisfies: some [x i , y] = 0.

Elliptic spaces

Bulletin of the American Mathematical Society, 1991

A simply connected CW complex of finite type is elliptic if it has finite L.S.-category and if it... more

Graded Lie algebras with finite polydepth

Annales Scientifiques de l’École Normale Supérieure, 2003

If A is a graded connected algebra then we define a new invariant, polydepth A, which is finite i... more If A is a graded connected algebra then we define a new invariant, polydepth A, which is finite if Ext * A (M, A) = 0 for some A-module M of at most polynomial growth. THEOREM 1: If f : X → Y is a continuous map of finite category, and if the orbits of H * (ΩY) acting in the homology of the homotopy fibre grow at most polynomially, then H * (ΩY) has finite polydepth. THEOREM 5: If L is a graded Lie algebra and polydepth UL is finite then either L is solvable and UL grows at most polynomially or else for some integer d and all r, k+d i=k+1 dim Li k r , k some k(r).

Rational Homotopy Theory via Sullivan Models: A Survey

Notices of the International Congress of Chinese Mathematicians

the last 40 years, and most especially to our good friend, Jean-Claude Thomas, frequently our co-... more the last 40 years, and most especially to our good friend, Jean-Claude Thomas, frequently our co-author, including in particular for the two volumes on rational homotopy. Contents 1 Notation and terminology 2 Sullivan models 3 The spatial realization of a Sullivan algebra 4 Homotopy groups 5 The homotopy Lie algebra and the fundamental group 6 Whitehead products and nilpotence 7 Hurewicz homomorphisms 8 Rationalizations, Malcev completions and Sullivan spaces 9 LS category 10 The trichotomy : growth of homotopy groups 11 Minimal ∧-models 12 Minimal ∧-models of a fibration 13 Holonomy representations 14 Fiber squares 15 Mapping spaces 16 Acyclic closures and loop spaces 17 Depth 18 Configuration spaces 19 The complement of arrangements 20 The group of homotopy self-equivalences of a space

Enriched Lie algebras in topology, I

To each path connected space X the Sullivan theory of minimal models associates a commutative dif... more To each path connected space X the Sullivan theory of minimal models associates a commutative differential graded algebra, its minimal model (∧V, d), and with it a graded Lie algebra LX that is the homotopy Lie algebra of its geometric realization < ∧V, d >. When X is a simply connected space with finite Betti numbers, then LX is isomorphic to the Lie algebra of X, π∗(ΩX)⊗Q, equipped with the Whitehead bracket. In the general case, LX is a complete enriched Lie algebra: An enriched Lie algebra is a graded Lie algebra L = L≥0 together with a family of ideals Iα indexed by a partially ordered set with α ≥ β if and only if Iα ⊂ Iβ, and such that each quotient L/Iα is a finite dimensional nilpotent Lie algebra. The enriched Lie algebra (L, {Iα}) is complete if L = lim ←−α L/Iα. Enriched Lie algebras are the natural extension of graded Lie algebras for path connected spaces, because each complete enriched Lie algebra L is the homotopy Lie algebra of a path connected space X. This t...

A note on Gorenstein spaces

Associated with an augmented differential graded algebra R= R^≥ 0 is a homotopy invariant T(R). T... more Associated with an augmented differential graded algebra R= R^≥ 0 is a homotopy invariant T(R). This is a graded vector space, and if H^0(R) is the ground field and H^>N(R)= 0 then dim T(R)= 1 if and only if H(R) is a Poincaré duality algebra. In the case of Sullivan extensions W→ W⊗ Z→ Z in which dim H( Z)<∞ we show that T( W⊗ Z)= T( W)⊗ T( Z). This is applied to finite dimensional CW complexes X where the fundamental group G acts nilpotently in the cohomology H(X; Q) of the universal covering space. If H(X; Q) is a Poincaré duality algebra and H(X; Q) and H(BG; Q) are finite dimensional then they are also Poincaré duality algebras.

Homotopy Lie algebras and Sullivan Lie algebras

Depth of a connected graded Lie algebra of finite type

Structure of a graded Lie algebra of finite depth

The depth of a Riemann surface and of a right-angled Artin group

Journal of Homotopy and Related Structures, 2019

We consider two families of spaces, X : the closed orientable Riemann surfaces of genus g > 0 and... more We consider two families of spaces, X : the closed orientable Riemann surfaces of genus g > 0 and the classifying spaces of right-angled Artin groups. In both cases we compare the depth of the fundamental group with the depth of an associated Lie algebra, L, that can be determined by the minimal Sullivan algebra. For these spaces we prove that depth Q[π 1 (X)] = depth L and give precise formulas for the depth.