DOCSLIB.ORG

Linear Algebraic Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Algebraic Groups Appendix A Linear Algebraic Groups A.1 Linear Algebraic Groups Let us fix an algebraically closed field K. We work in the category of affine varieties over K. The purpose of this section is to give a brief exposition on the basic facts of algebraic groups. Definition A.1.1 A linear algebraic group is an affine algebraic variety G together with a unit element e 2 G and morphisms m W G G ! G and i W G ! G satisfying the group axioms (a) m.; e/ D m.e;/D for all 2 G; (b) m.; i.// D m.i./; / D e for all 2 G; (c) m.; m.; // D m.m.; /; / (associativity) for all ; ; 2 G. Often we will just write and 1 instead of m.; / and i./, respectively. Example A.1.2 The group GLn, the set of invertible nn matrices over K, is a linear algebraic group. Any Zariski closed subgroup G GLn is a linear algebraic group. In fact, every linear algebraic group is isomorphic to a Zariski closed subgroup of GLn (see Borel [1, Proposition I.1.10]). This justifies calling it a linear algebraic group. G The axioms for a linear algebraic group can also be stated in terms of the coordinate ring. The maps m W G G ! G and i W G ! G correspond to ring homomorphisms m W KŒG ! KŒG ˝ KŒG and i W KŒG ! KŒG. The unit element e 2 G corresponds to a ring homomorphism W KŒG ! K defined by f 7! f .e/. The axioms for a linear algebraic group translate to (a) .id ˝/ ı m D . ˝ id/ ı m D id; (b) .i ˝ id/ ı m D .id ˝i/ ı m D (using the inclusion K KŒG we can view W KŒG ! K as a map KŒG ! KŒG); (c) .id ˝m/ ı m D .m ˝ id/ ı m. © Springer-Verlag Berlin Heidelberg 2015 297 H. Derksen, G. Kemper, Computational Invariant Theory, Encyclopaedia of Mathematical Sciences 130, DOI 10.1007/978-3-662-48422-7 298 A Linear Algebraic Groups Example A.1.3 The additive group Ga is defined as the additive group of K.We define m.x; y/ D x C y, i.x/ Dx for all x; y 2 K and e D 0 2 K. The coordinate ring KŒGa is equal to the polynomial ring KŒt. The homomorphism m W KŒt ! KŒt ˝ KŒt is defined by m.t/ D t ˝ 1 C 1 ˝ t, i W KŒt ! KŒt is defined by t 7!t and W KŒt ! K is defined by f 7! f .0/. Example A.1.4 The group GLn is an algebraic group. The coordinate ring KŒGLn is isomorphic to KŒfzi;j j 1 Ä i; j Ä ng; T=. f /; . / 1 . /n Œ Œ where f D T det Z with ZPthe matrix zi;j i;jD1.Nowm W K GLn ! K GLn ˝ Œ n ; . // K GLn is defined by zi;j 7! kD1 zi;k ˝ zk;j for all i j. Notice that m det Z D det.Z/ ˝ det.Z/. Example A.1.5 Let T D .K/r. The coordinate ring KŒT is isomorphic to Œ ;:::; ; 1;:::; 1: K z1 zr z1 zr The ring homomorphisms m W KŒT ! KŒT ˝ KŒT, i W KŒT ! KŒT and Œ . / . / 1 . / 1 W K T ! K are given by m zi D zi ˝ zi, i zi D zi and zi D for all i. G Definition A.1.6 Suppose that X is an affine variety. A regular action of G on X is a morphism W G X ! X satisfying the axioms for an action (a) .e; x/ D x for all x 2 X; (b) .; .; x// D .m.; /; x/ for all ; 2 G and all x 2 X. We will often just write x instead of .; x/. Instead of saying that G acts regularly on X, we often just say that X is a G-variety. Definition A.1.7 Suppose that V is a (possibly infinite dimensional) vector space over K. A linear action W G V ! V of G on a vector spaceP is called rational ? Œ .; v/ l v ./ if there isP a map W V ! V ˝ K G satisfying D iD1 i fi whenever ?.v/ l v D iD1 i ˝ fi Lemma A.1.8 Suppose that a linear action of a linear algebraic group G on a vector space V is rational. For every v 2 V there exists a finite dimensional G-stable subspace W  V containing v. The restriction of the G-action to W is also rational. P ?.v/ l v ;:::; Proof Write D iD1 i ˝ fi with f1 fl linearly independent over K.Let W D K.G v/, the linear span of the orbit G v. It is not hard to see that W is a subspace of the space spanned by v1;:::;vl. This shows that dim W < 1. We can restrict ? to W to obtain a map W ! W ˝KŒG. Therefore the G-action on W is rational. ut A.2 The Lie Algebra of a Linear Algebraic Group 299 The content of Lemma A.1.8 can be expressed by saying that a rational action is “locally finite”. If G acts regularly on an affine variety X,thenG also acts on the coordinate ring KŒX.For 2 G and f 2 KŒX we define f 2 KŒX by . f /.x/ D f . 1 x/ for all x 2 X. AmapX ! Y between two sets on which G acts is called G-equivariant if '. x/ D '.x/ for all x 2 X and all 2 G. Lemma A.1.9 Suppose that X is a G-variety. Then there exists a finite dimensional rational representation V and a G-equivariant closed embedding X ,! V. Proof Choose generators f1;:::;fr of KŒX and let W be a finite dimensional G- stable subspace of KŒX containing f1;:::;fr. The inclusion W ! KŒX extends to a surjective G-equivariant ring homomorphism S.W/ KŒX,whereS.W/ is the symmetric algebra. This ring homomorphism corresponds to a G-equivariant closed embedding X ,! W where W is the dual space of W. ut The following two propositions can be found in Borel [1, Proposition I.1.2]. Proposition A.1.10 A linear algebraic group G is smooth. Proposition A.1.11 Let Gı be the connected component of e 2 G. Then Gı is a normal subgroup of G of finite index. A.2 The Lie Algebra of a Linear Algebraic Group We will first study the dual vector space KŒG of KŒG. As we will see, KŒG contains the Lie algebra g as a finite dimensional subspace. For every 2 G we can define W KŒG ! K by f 7! f ./. In this way we can view G as a subset of KŒG. By Definition A.2.1 below, the group structure of G equips KŒG with an associative algebra structure with identity element D e. This even allows us to see KŒG as an enveloping algebra of g (containing the universal enveloping algebra of g). Our approach is similar to Borel [1,§I.3]. Definition A.2.1 Suppose that ı; 2 KŒG. Then we define the convolution ı 2 KŒG as the composition of ˝ ı W KŒG ˝ KŒG ! K ˝ K D KPand m W Œ Œ Œ Œ . / K G ! K G ˝ K G . In other words, suppose f 2 K G and m f D i gi ˝ hi, then X . ı/.f / D .gi/ı.hi/: i 300 A Linear Algebraic Groups Proposition A.2.2 ThespaceKŒG is an associative algebra with the multiplica- tion and unit element D e. Proof From the axiom m.m.; /; / D m.; m.; //; it follows that .m ˝ id/ ı m D .id ˝m/ ı m: This multiplication is associative, because .ı / ' D ...ı ˝ /ı m/ ˝ '/ ı m D .ı ˝ ˝ '/ ı .m ˝ id/ ı m D D .ı ˝ ˝ '/ ı .id ˝m/ ı m D .ı ˝ .. ˝ '/ ı m// ı m D ı . '/: (A.2.1) Notice that from m.e;/D it follows that . ˝ id/ ı m D id. We get ı D .ı ˝ / ı m D ı ı .id ˝/ ı m D ı ı id D ı; and similarly ı D ı. ut Example A.2.3 If f 2 KŒG and 2 G,wedefineL . f / and R . f / by L . f /./ D f ./ and R . f /./ D f ./. Notice that .id ˝ / ı m D R and . ˝id/ ı m D L . From this it follows that ı D ı ı .id ˝ / ı m D ı ı R and ı D ı ı . ˝ id/ ı m D ı ı L : G Definition A.2.4 We define the Lie algebra g of G as the set of all ı 2 KŒG satisfying ı. fg/ D ı. f /g.e/ C f .e/ı.g/: (A.2.2) for all f ; g 2 KŒG. An element ı 2 KŒG satisfying (A.2.2) is called a point derivation of KŒG at e.Letme be the maximal ideal of KŒG vanishing at e 2 G. The Zariski tangent . / . = 2/ = 2 space Te G of G at e is defined as me me which is the dual space of me me.If A.2 The Lie Algebra of a Linear Algebraic Group 301 ı . = 2/ .ı/ Œ 2 me me , we can define 2 K G by .ı/ ı. / 2/: f D f f e C me It is not hard to show that .ı/ is a point derivation at e and that ı 7! .ı/ is an isomorphism between Te.G/ and g.
Load more

Recommended publications
  • Classification of Finite Abelian Groups
    Math 317 C1 John Sullivan Spring 2003 Classification of Finite Abelian Groups (Notes based on an article by Navarro in the Amer. Math. Monthly, February 2003.) The fundamental theorem of finite abelian groups expresses any such group as a product of cyclic groups: Theorem. Suppose G is a finite abelian group. Then G is (in a unique way) a direct product of cyclic groups of order pk with p prime. Our first step will be a special case of Cauchy’s Theorem, which we will prove later for arbitrary groups: whenever p |G| then G has an element of order p. Theorem (Cauchy). If G is a finite group, and p |G| is a prime, then G has an element of order p (or, equivalently, a subgroup of order p). ∼ Proof when G is abelian. First note that if |G| is prime, then G = Zp and we are done. In general, we work by induction. If G has no nontrivial proper subgroups, it must be a prime cyclic group, the case we’ve already handled. So we can suppose there is a nontrivial subgroup H smaller than G. Either p |H| or p |G/H|. In the first case, by induction, H has an element of order p which is also order p in G so we’re done. In the second case, if ∼ g + H has order p in G/H then |g + H| |g|, so hgi = Zkp for some k, and then kg ∈ G has order p. Note that we write our abelian groups additively. Definition. Given a prime p, a p-group is a group in which every element has order pk for some k.
    [Show full text]
  • A NEW APPROACH to RANK ONE LINEAR ALGEBRAIC GROUPS An
    A NEW APPROACH TO RANK ONE LINEAR ALGEBRAIC GROUPS DANIEL ALLCOCK Abstract. One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps of the standard development, including the self-normalizing property of Borel subgroups. An awkwardness of the theory of linear algebraic groups is that one must develop a lot of material before one can even characterize PGL2. Our goal here is to show how to develop the root system, etc., using only the completeness of the flag variety, its immediate consequences, and some facts about solvable groups. In particular, one can skip over the usual analysis of Cartan subgroups, the fact that G is the union of its Borel subgroups, the connectedness of torus centralizers, and the normalizer theorem (that Borel subgroups are self-normalizing). The main idea is a new approach to the structure of rank 1 groups; the key step is lemma 5. All algebraic geometry is over a fixed algebraically closed field. G always denotes a connected linear algebraic group with Lie algebra g, T a maximal torus, and B a Borel subgroup containing it. We assume the structure theory for connected solvable groups, and the completeness of the flag variety G/B and some of its consequences. Namely: that all Borel subgroups (resp. maximal tori) are conjugate; that G is nilpotent if one of its Borel subgroups is; that CG(T )0 lies in every Borel subgroup containing T ; and that NG(B) contains B of finite index and (therefore) is self-normalizing.
    [Show full text]
  • The General Linear Group
    18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices.
    [Show full text]
  • Arxiv:1906.01274V1 [Math.NT] 4 Jun 2019 Rpsto .] Aqe Ishmefas Rvdsoepofi I Yale 1.4.1]
    CHOW’S THEOREM FOR SEMI-ABELIAN VARIETIES AND BOUNDS FOR SPLITTING FIELDS OF ALGEBRAIC TORI CHIA-FU YU Abstract. A theorem of Chow concerns homomorphisms of two abelian vari- eties under a primary field extension base change. In this paper we generalize Chow’s theorem to semi-abelian varieties. This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori. 1. Introduction Let k be a field, k¯ an algebraic closure of k, and ks the separable closure of k in k¯. A connected algebraic k-group T is an algebraic torus if there is a k¯-isomorphism d T ⊗k k¯ ≃ (Gm) ⊗k k of algebraic groups for some integer d ≥ 0. We say T splits d over a field extension K of k if there is a K-isomorphism T ⊗k K ≃ (Gm) ⊗k K. This paper is motivated from the following fundamental result. Theorem 1.1. Any algebraic k-torus T splits over ks. In other words, T splits over a finite separable field extension of k. This theorem is well known and it is stated and proved in the literature several times. Surprisingly, different authors choose their favorite proofs which are all quite different. As far as we know, the first proof is given by Takashi Ono [40, Proposition 1.2.1]. Armand Borel gives a different proof in his book Linear Algebraic Groups; see [3, Proposition 8.11]. In the second edition of his book Linear Algebraic Groups [49], T.A.
    [Show full text]
  • Unitary Group - Wikipedia
    Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group Unitary group In mathematics, the unitary group of degree n, denoted U( n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL( n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group. In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group. The unitary group U( n) is a real Lie group of dimension n2. The Lie algebra of U( n) consists of n × n skew-Hermitian matrices, with the Lie bracket given by the commutator. The general unitary group (also called the group of unitary similitudes ) consists of all matrices A such that A∗A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix. Contents Properties Topology Related groups 2-out-of-3 property Special unitary and projective unitary groups G-structure: almost Hermitian Generalizations Indefinite forms Finite fields Degree-2 separable algebras Algebraic groups Unitary group of a quadratic module Polynomial invariants Classifying space See also Notes References Properties Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group 1 of 7 2/23/2018, 10:13 AM Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group homomorphism The kernel of this homomorphism is the set of unitary matrices with determinant 1.
    [Show full text]
  • Material on Algebraic and Lie Groups
    2 Lie groups and algebraic groups. 2.1 Basic Definitions. In this subsection we will introduce the class of groups to be studied. We first recall that a Lie group is a group that is also a differentiable manifold 1 and multiplication (x, y xy) and inverse (x x ) are C1 maps. An algebraic group is a group7! that is also an algebraic7! variety such that multi- plication and inverse are morphisms. Before we can introduce our main characters we first consider GL(n, C) as an affi ne algebraic group. Here Mn(C) denotes the space of n n matrices and GL(n, C) = g Mn(C) det(g) =) . Now Mn(C) is given the structure nf2 2 j 6 g of affi ne space C with the coordinates xij for X = [xij] . This implies that GL(n, C) is Z-open and as a variety is isomorphic with the affi ne variety 1 Mn(C) det . This implies that (GL(n, C)) = C[xij, det ]. f g O Lemma 1 If G is an algebraic group over an algebraically closed field, F , then every point in G is smooth. Proof. Let Lg : G G be given by Lgx = gx. Then Lg is an isomorphism ! 1 1 of G as an algebraic variety (Lg = Lg ). Since isomorphisms preserve the set of smooth points we see that if x G is smooth so is every element of Gx = G. 2 Proposition 2 If G is an algebraic group over an algebraically closed field F then the Z-connected components Proof.
    [Show full text]
  • Symmetries on the Lattice
    Symmetries On The Lattice K.Demmouche January 8, 2006 Contents Background, character theory of finite groups The cubic group on the lattice Oh Representation of Oh on Wilson loops Double group 2O and spinor Construction of operator on the lattice MOTIVATION Spectrum of non-Abelian lattice gauge theories ? Create gauge invariant spin j states on the lattice Irreducible operators Monte Carlo calculations Extract masses from time slice correlations Character theory of point groups Groups, Axioms A set G = {a, b, c, . } A1 : Multiplication ◦ : G × G → G. A2 : Associativity a, b, c ∈ G ,(a ◦ b) ◦ c = a ◦ (b ◦ c). A3 : Identity e ∈ G , a ◦ e = e ◦ a = a for all a ∈ G. −1 −1 −1 A4 : Inverse, a ∈ G there exists a ∈ G , a ◦ a = a ◦ a = e. Groups with finite number of elements → the order of the group G : nG. The point group C3v The point group C3v (Symmetry group of molecule NH3) c ¡¡AA ¡ A ¡ A Z ¡ A Z¡ A Z O ¡ Z A ¡ Z A ¡ Z A Z ¡ Z A ¡ ZA a¡ ZA b G = {Ra(π),Rb(π),Rc(π),E(2π),R~n(2π/3),R~n(−2π/3)} noted G = {A, B, C, E, D, F } respectively. Structure of Groups Subgroups: Definition A subset H of a group G that is itself a group with the same multiplication operation as G is called a subgroup of G. Example: a subgroup of C3v is the subset E, D, F Classes: Definition An element g0 of a group G is said to be ”conjugate” to another element g of G if there exists an element h of G such that g0 = hgh−1 Example: on can check that B = DCD−1 Conjugacy Class Definition A class of a group G is a set of mutually conjugate elements of G.
    [Show full text]
  • 1 Affine Varieties
    1 Affine Varieties We will begin following Kempf's Algebraic Varieties, and eventually will do things more like in Hartshorne. We will also use various sources for commutative algebra. What is algebraic geometry? Classically, it is the study of the zero sets of polynomials. We will now fix some notation. k will be some fixed algebraically closed field, any ring is commutative with identity, ring homomorphisms preserve identity, and a k-algebra is a ring R which contains k (i.e., we have a ring homomorphism ι : k ! R). P ⊆ R an ideal is prime iff R=P is an integral domain. Algebraic Sets n n We define affine n-space, A = k = f(a1; : : : ; an): ai 2 kg. n Any f = f(x1; : : : ; xn) 2 k[x1; : : : ; xn] defines a function f : A ! k : (a1; : : : ; an) 7! f(a1; : : : ; an). Exercise If f; g 2 k[x1; : : : ; xn] define the same function then f = g as polynomials. Definition 1.1 (Algebraic Sets). Let S ⊆ k[x1; : : : ; xn] be any subset. Then V (S) = fa 2 An : f(a) = 0 for all f 2 Sg. A subset of An is called algebraic if it is of this form. e.g., a point f(a1; : : : ; an)g = V (x1 − a1; : : : ; xn − an). Exercises 1. I = (S) is the ideal generated by S. Then V (S) = V (I). 2. I ⊆ J ) V (J) ⊆ V (I). P 3. V ([αIα) = V ( Iα) = \V (Iα). 4. V (I \ J) = V (I · J) = V (I) [ V (J). Definition 1.2 (Zariski Topology). We can define a topology on An by defining the closed subsets to be the algebraic subsets.
    [Show full text]
  • Matrix Lie Groups
    Maths Seminar 2007 MATRIX LIE GROUPS Claudiu C Remsing Dept of Mathematics (Pure and Applied) Rhodes University Grahamstown 6140 26 September 2007 RhodesUniv CCR 0 Maths Seminar 2007 TALK OUTLINE 1. What is a matrix Lie group ? 2. Matrices revisited. 3. Examples of matrix Lie groups. 4. Matrix Lie algebras. 5. A glimpse at elementary Lie theory. 6. Life beyond elementary Lie theory. RhodesUniv CCR 1 Maths Seminar 2007 1. What is a matrix Lie group ? Matrix Lie groups are groups of invertible • matrices that have desirable geometric features. So matrix Lie groups are simultaneously algebraic and geometric objects. Matrix Lie groups naturally arise in • – geometry (classical, algebraic, differential) – complex analyis – differential equations – Fourier analysis – algebra (group theory, ring theory) – number theory – combinatorics. RhodesUniv CCR 2 Maths Seminar 2007 Matrix Lie groups are encountered in many • applications in – physics (geometric mechanics, quantum con- trol) – engineering (motion control, robotics) – computational chemistry (molecular mo- tion) – computer science (computer animation, computer vision, quantum computation). “It turns out that matrix [Lie] groups • pop up in virtually any investigation of objects with symmetries, such as molecules in chemistry, particles in physics, and projective spaces in geometry”. (K. Tapp, 2005) RhodesUniv CCR 3 Maths Seminar 2007 EXAMPLE 1 : The Euclidean group E (2). • E (2) = F : R2 R2 F is an isometry . → | n o The vector space R2 is equipped with the standard Euclidean structure (the “dot product”) x y = x y + x y (x, y R2), • 1 1 2 2 ∈ hence with the Euclidean distance d (x, y) = (y x) (y x) (x, y R2).
    [Show full text]
  • Lie Algebras of Generalized Quaternion Groups 1 Introduction 2
    Lie Algebras of Generalized Quaternion Groups Samantha Clapp Advisor: Dr. Brandon Samples Abstract Every finite group has an associated Lie algebra. Its Lie algebra can be viewed as a subspace of the group algebra with certain bracket conditions imposed on the elements. If one calculates the character table for a finite group, the structure of its associated Lie algebra can be described. In this work, we consider the family of generalized quaternion groups and describe its associated Lie algebra structure completely. 1 Introduction The Lie algebra of a group is a useful tool because it is a vector space where linear algebra is available. It is interesting to consider the Lie algebra structure associated to a specific group or family of groups. A Lie algebra is simple if its dimension is at least two and it only has f0g and itself as ideals. Some examples of simple algebras are the classical Lie algebras: sl(n), sp(n) and o(n) as well as the five exceptional finite dimensional simple Lie algebras. A direct sum of simple lie algebras is called a semi-simple Lie algebra. Therefore, it is also interesting to consider if the Lie algebra structure associated with a particular group is simple or semi-simple. In fact, the Lie algebra structure of a finite group is well known and given by a theorem of Cohen and Taylor [1]. In this theorem, they specifically describe the Lie algebra structure using character theory. That is, the associated Lie algebra structure of a finite group can be described if one calculates the character table for the finite group.
    [Show full text]
  • Some Finiteness Results for Groups of Automorphisms of Manifolds 3
    SOME FINITENESS RESULTS FOR GROUPS OF AUTOMORPHISMS OF MANIFOLDS ALEXANDER KUPERS Abstract. We prove that in dimension =6 4, 5, 7 the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of Rn and various types of automorphisms of 2-connected manifolds. Contents 1. Introduction 1 2. Homologically and homotopically finite type spaces 4 3. Self-embeddings 11 4. The Weiss fiber sequence 19 5. Proofs of main results 29 References 38 1. Introduction Inspired by work of Weiss on Pontryagin classes of topological manifolds [Wei15], we use several recent advances in the study of high-dimensional manifolds to prove a structural result about diffeomorphism groups. We prove the classifying spaces of such groups are often “small” in one of the following two algebro-topological senses: Definition 1.1. Let X be a path-connected space. arXiv:1612.09475v3 [math.AT] 1 Sep 2019 · X is said to be of homologically finite type if for all Z[π1(X)]-modules M that are finitely generated as abelian groups, H∗(X; M) is finitely generated in each degree. · X is said to be of finite type if π1(X) is finite and πi(X) is finitely generated for i ≥ 2. Being of finite type implies being of homologically finite type, see Lemma 2.15. Date: September 4, 2019. Alexander Kupers was partially supported by a William R.
    [Show full text]
  • GEOMETRY and GROUPS These Notes Are to Remind You of The
    GEOMETRY AND GROUPS These notes are to remind you of the results from earlier courses that we will need at some point in this course. The exercises are entirely optional, although they will all be useful later in the course. Asterisks indicate that they are harder. 0.1 Metric Spaces (Metric and Topological Spaces) A metric on a set X is a map d : X × X → [0, ∞) that satisfies: (a) d(x, y) > 0 with equality if and only if x = y; (b) Symmetry: d(x, y) = d(y, x) for all x, y ∈ X; (c) Triangle Rule: d(x, y) + d(y, z) > d(x, z) for all x, y, z ∈ X. A set X with a metric d is called a metric space. For example, the Euclidean metric on RN is given by d(x, y) = ||x − y|| where v u N ! u X 2 ||a|| = t |an| n=1 is the norm of a vector a. This metric makes RN into a metric space and any subset of it is also a metric space. A sequence in X is a map N → X; n 7→ xn. We often denote this sequence by (xn). This sequence converges to a limit ` ∈ X when d(xn, `) → 0 as n → ∞ . A subsequence of the sequence (xn) is given by taking only some of the terms in the sequence. So, a subsequence of the sequence (xn) is given by n 7→ xk(n) where k : N → N is a strictly increasing function. A metric space X is (sequentially) compact if every sequence from X has a subsequence that converges to a point of X.
    [Show full text]