DOCSLIB.ORG

Introduction to Quillen's Algebraic K-Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Algebraic K-theory Daniel Ramras Typed by Virgil Chan 29 October 2004 { 11 November 2004 2 Contents Preface 5 1 Projective Modules7 1.1 Equivalent Definitions..............................7 1.2 Topological K-theory...............................7 1.3 K0 for Schemes..................................8 2 Higher K-theory9 2.1 Simplicial Sets...................................9 2.1.1 Canonical Example{Singular Sets.................... 10 2.1.2 Formal Definition............................. 11 2.1.3 Geometric Realisation of a Simplicial Set................ 11 2.2 Classifying Spaces of a Small Category..................... 12 2.3 Definition of Higher K-theory.......................... 14 2.3.1 The Q-construction............................ 14 2.3.2 Exact Categories and the Q-construction................ 16 2.3.3 Universal Property of the Q-construction................ 18 2.3.4 An Explicit Map α : K0(R) ! π1 Q P ............... 23 2.3.5 Another Approach to π1 Q M .................... 26 3 Bloch's Formula for Chow Groups 31 3.1 Product...................................... 35 4 Theorem A and D´evissage 39 3 4 CONTENTS Preface These are notes from a series of three lectures I gave at Stanford in Fall 2004. They cover ∼ Swan's Theorem, the Q-construction, the isomorphism π1 Q P (R) = K0(R), Bloch's Formula, and D´evissage. Thanks to Virgil Chan for turning my handwritten notes into a readable document. 5 6 CONTENTS Chapter 1 Projective Modules 1.1 Equivalent Definitions Let R be a (commutative) ring with identity 1 = 1R; M be a finitely generated R-module. We say M is projective if it is a direct summand of a free module. We denote P (R) the category of projective R-modules with module homomorphisms, then the Grothendieck group K0(R) is defined as follows: Let F be the free abelain group with a generator hP i for each isomorphism class of projective modules over R, and let R be the subgroup of F generated by hP ⊕ Qi − hP i − hQi : (Note: P ⊕ Q 2 P (R) whenever P , Q 2 P (R)). We define K0(R) := F=R and write [P ] for hP i R. In other words, we formally add inverse to the monoid M of isomorphism classes. K0(R) is universal in the sense that any monoid map M ! A with A an abelian group factors through K0(R) uniquely. Remark [P ] = [Q] if and only if P is stably isomorphic to Q, that is, there exists n such n ∼ n that P ⊕ R = Q ⊕ R . So K0(R) is the "group of modules up to stable isomorphism". ∼ Example 1.1.1. If R is a field, then K0(R) = Z, generated by [R]. 1.2 Topological K-theory Let X be a compact Hausdorff space. We form a group K0(X) just as before, replacing projective modules by vector bundles over X. (We can use complex, real, or quaternion, but 0 0 n ∼ 0 n let us use C.) Again, in K (X) we have [V ] = [V ] if and only if V ⊕ " = V ⊕ " for some n n n, where " is the n-dimensional trivial bundle X × C ! X. 7 8 CHAPTER 1. PROJECTIVE MODULES 0 ∼ Theorem 1.2.1 ([Swa62]). K (X) = K0(C(X)), where C(X) is the ring of continuous functions X ! C. Sketch of Proof. We need to show that vector bundles correspond to projective C(X)-modules: fV ! Xg ! Γ(V ) := fcontinuous sections of V g : If V is a vector bundle, then there exists W such that V ⊕ W = "n, so Γ(V ) ⊕ Γ(W ) = Γ("n) = C(X)n. i.e., Γ is projective. If M is a projective, finitely generated C(X)-module, then M ⊕ P ∼= C(X)n for some C(X)-module P and some integer n. So there exists f : C(X)n ! C(X)n with f 2 = f. (f(C(X)) = M.) Now f corresponds to a map g : "n ! "n with g2 = g, and we can check that im (g) is a vector bundle with sections Γ(im (g)) = M. (To get g, note Γ(ξ)=fs 2 Γ j s(X) = 0g is fibre over X. So f induces map on each fibre, which is a bundle map.) 1.3 K0 for Schemes Let X be a N¨otherianscheme. We define K0(X) to be the Grothendieck group of the category of coherent sheaves on X (sheaves which are locally finitely generated modules). This is the group with generators [F] for each isomorphism classes of coherent sheaves on X, and relations [F] = [F 0] + [F 00] for each short exact sequence 0 !F 0 !F!F 00 ! 0: Note we are not assume such sequences split in the category of coherent sheaves, by in K-theory, [F] = [F 0] + [F 00] = [F 0 ⊕ F 00] for each short exact sequence 0 !F 0 !F!F 00 ! 0: This is the case originally studied by Grothendieck in his proof of the Riemann-Roch Theo- rem. Chapter 2 Higher K-theory For a commutative ring R with identity, we define ab K1(R) := GL (1;R) ab = colim GL (n; R) ; n K2(R) := group of non-trivial relations among elementary matrices. λ Recall an elementary matrix eij 2 GL (n; Λ) has 1's on the main diagonal, λ in the (i; j)- entry, and 0's else where. The groups K0, K1, and K2 are related by exact sequences. For example, if Λ is a Dedekind domain with fraction field F , there is a localisation sequence: M M K2(F ) K1(Λ=p) K1(Λ) K1(F ) K0(Λ=p) K0(Λ) K0(F ) 0 p p Quillen extended this to the left by defining higher K-groups (even to odd other K2-terms one needs a completely new way of thinking about K0, K1, K2). 2.1 Simplicial Sets A simplicial set X• provides a combinatorial description of a topological space. It is a sequence of sets X0;X1;X2; ···, and structure maps di : Xn ! Xn−1; si : Xn ! Xn+1 for each i with 0 ≤ i ≤ n, satisfying the following relations 9 10 CHAPTER 2. HIGHER K-THEORY didj = dj−1di if i < j; disj = sj−1di if i < j; djsj = dj+1sj = id; (2.1.1) disj = sjdi−1 if i > j + 1; sisj = sj+1si if i ≤ j: We think of Xi as the collection of i-simplices of X•, with face di and degeneracy si operators. For example, given an \edge" e 2 X1, its \endpoints" are the \vertices" d0(e); d1(e) 2 X0. si's correspond to repeating a vertex in a simplex. For instance, if v is a vertex, then (v; v) is a degenerate 1-simplex. More formally, if σ 2 Xn is a simplex with an ordered set of vertices σ = [v0; v1; ··· ; vn], then di(σ) = [v0; v1; ··· ; v^i; ··· ; vn] 2 Xn−1; and si(σ) = [v0; ··· ; vi; vi; ··· ; vn] 2 Xn+1: From this readers can verify Equation (2.1.1) easily. 2.1.1 Canonical Example{Singular Sets Let X be any topological space, we define the singular set of X to be the simplicial set n Sn(X) := ff : ∆ ! X j f is continuousg ; ( n ) n n+1 X where ∆ := (x0; ··· ; xn) 2 R 0 ≤ xi ≤ 1; xi = 1 is the standard n-simplex. To i=0 define face and degeneracy operators, we first define n−1 n δi : ∆ ! ∆ (inclusion as i-th face) ; n+1 n σi : ∆ ! ∆ (collapse onto i-th face) by δi(x0; ··· ; xn−1) = (x0; ··· ; xi−1; 0; ··· ; xn−1) σi(x0; ··· ; xn+1) = (x0; ··· ; xi + xi+1; ··· ; xn+1): Then define di(f) := f ◦ δi; si(f) := f ◦ σi: 2.1. SIMPLICIAL SETS 11 2.1.2 Formal Definition op op A simplicial set is a functor X• : ∆ ! Sets, where ∆ is the category of finite totally ordered sets and order-preserving morphisms. So if [n] := f0 < 1 < ··· < ng, then Xn := X•([n]) is the set of n-simplices, and the face and degeneracy faces are induced by the maps δi :[n] ! [n + 1] σi :[n] ! [n − 1] j if j < i; j if j ≤ i; δ (j) = (skip i) σ (j) = (repeat i) i j + 1 if j ≥ i i j − 1 if j > i Example 2.1.1 (The Simplicial Interval). The simplicial interval I• := ∆[1] is the simplicial set I0 = f0; 1g ; I1 = f[0; 1]; [0; 0]; [1; 1]g ; I2 = f[0; 0; 1]; [0; 1; 1]g ; with d0([0; 1]) = 1; d1([0; 1]) = 0 and et cetera. For a formal definition of I•, see Example 2.2.2. 2.1.3 Geometric Realisation of a Simplicial Set Given a simpicial set X•, we want to create a topological space ! G n jX•j := Xn × ∆ =∼ : n≥0 The relation \∼" identifies faces and collapses degeneracies: n (τ; δi(v)) ∼ (di(τ); v) for τ 2 Xn+1 and v 2 ∆ ; n (τ; σi(v)) ∼ (si(τ); v) for τ 2 Xn−1 and v 2 ∆ : 12 CHAPTER 2. HIGHER K-THEORY Theorem 2.1.2 (Milnor). (i)[Mil57, Theorem 1] jX•j is a CW-complex with one n-cell for each non-degenerate simplex τ 2 Xn. (ii)[Mil57, Theorem 2] jX• × Y•j and jX•j × jY•j are weakly equivalent, and are home- omorphic when jX•j or jY•j is locally finite. Here, X• × Y• has n-simplices Xn × Yn and si, di operate coordinate-wise. Remark 2.1.3.1. Earlier definitions of simplicial sets ignored degeneracies and hence we did not get jX• × Y•j = jX•j × jY•j. Note that (σ; τ) 2 Xn × Yn is degenerate only when σ and τ are both degenerate.
Recommended publications
  • Co-Occurrence Simplicial Complexes in Mathematics: Identifying the Holes of Knowledge Arxiv:1803.04410V1 [Physics.Soc-Ph] 11 M

    Co-Occurrence Simplicial Complexes in Mathematics: Identifying the Holes of Knowledge Arxiv:1803.04410V1 [Physics.Soc-Ph] 11 M

    Co-occurrence simplicial complexes in mathematics: identifying the holes of knowledge Vsevolod Salnikov∗, NaXys, Universit´ede Namur, 5000 Namur, Belgium ∗Corresponding author: [email protected] Daniele Cassese NaXys, Universit´ede Namur, 5000 Namur, Belgium ICTEAM, Universit´eCatholique de Louvain, 1348 Louvain-la-Neuve, Belgium Mathematical Institute, University of Oxford, OX2 6GG Oxford, UK [email protected] Renaud Lambiotte Mathematical Institute, University of Oxford, OX2 6GG Oxford, UK [email protected] and Nick S. Jones Department of Mathematics, Imperial College, SW7 2AZ London, UK [email protected] Abstract In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose for the first time a simplicial complex approach to word co-occurrences, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological meth- ods we explore the conceptual landscape of mathematical research, focusing on homological holes, regions with low connectivity in the simplicial structure. We find that homological holes are ubiquitous, which suggests that they capture some essential feature of research arXiv:1803.04410v1 [physics.soc-ph] 11 Mar 2018 practice in mathematics. Holes die when a subset of their concepts appear in the same ar- ticle, hence their death may be a sign of the creation of new knowledge, as we show with some examples. We find a positive relation between the dimension of a hole and the time it takes to be closed: larger holes may represent potential for important advances in the field because they separate conceptually distant areas.
  • Simplicial Complexes

    Simplicial Complexes

    46 III Complexes III.1 Simplicial Complexes There are many ways to represent a topological space, one being a collection of simplices that are glued to each other in a structured manner. Such a collection can easily grow large but all its elements are simple. This is not so convenient for hand-calculations but close to ideal for computer implementations. In this book, we use simplicial complexes as the primary representation of topology. Rd k Simplices. Let u0; u1; : : : ; uk be points in . A point x = i=0 λiui is an affine combination of the ui if the λi sum to 1. The affine hull is the set of affine combinations. It is a k-plane if the k + 1 points are affinely Pindependent by which we mean that any two affine combinations, x = λiui and y = µiui, are the same iff λi = µi for all i. The k + 1 points are affinely independent iff P d P the k vectors ui − u0, for 1 ≤ i ≤ k, are linearly independent. In R we can have at most d linearly independent vectors and therefore at most d+1 affinely independent points. An affine combination x = λiui is a convex combination if all λi are non- negative. The convex hull is the set of convex combinations. A k-simplex is the P convex hull of k + 1 affinely independent points, σ = conv fu0; u1; : : : ; ukg. We sometimes say the ui span σ. Its dimension is dim σ = k. We use special names of the first few dimensions, vertex for 0-simplex, edge for 1-simplex, triangle for 2-simplex, and tetrahedron for 3-simplex; see Figure III.1.
  • Full Text (PDF Format)

    Full Text (PDF Format)

    ASIAN J. MATH. © 1997 International Press Vol. 1, No. 2, pp. 330-417, June 1997 009 K-THEORY FOR TRIANGULATED CATEGORIES 1(A): HOMOLOGICAL FUNCTORS * AMNON NEEMANt 0. Introduction. We should perhaps begin by reminding the reader briefly of Quillen's Q-construction on exact categories. DEFINITION 0.1. Let £ be an exact category. The category Q(£) is defined as follows. 0.1.1. The objects of Q(£) are the objects of £. 0.1.2. The morphisms X •^ X' in Q(£) between X,X' G Ob{Q{£)) = Ob{£) are isomorphism classes of diagrams of morphisms in £ X X' \ S Y where the morphism X —> Y is an admissible mono, while X1 —y Y is an admissible epi. Perhaps a more classical way to say this is that X is a subquotient of X''. 0.1.3. Composition is defined by composing subquotients; X X' X' X" \ ^/ and \ y/ Y Y' compose to give X X' X" \ s \ s Y PO Y' \ S Z where the square marked PO is a pushout square. The category Q{£) can be realised to give a space, which we freely confuse with the category. The Quillen if-theory of the exact category £ was defined, in [9], to be the homotopy of the loop space of Q{£). That is, Ki{£)=ni+l[Q{£)]. Quillen proved many nice functoriality properties for his iT-theory, and the one most relevant to this article is the resolution theorem. The resolution theorem asserts the following. 7 THEOREM 0.2. Let F : £ —> J be a fully faithful, exact inclusion of exact categories.
  • Higher Algebraic K-Theory I

    Higher Algebraic K-Theory I

    1 Higher algebraic ~theory: I , * ,; Daniel Quillen , ;,'. ··The·purpose of..thispaper.. is.to..... develop.a.higher. X..,theory. fpJ;' EiddUiy!!. categQtl~ ... __ with euct sequences which extends the ell:isting theory of ths Grothsndieck group in a natural wll7. To describe' the approach taken here, let 10\ be an additive category = embedded as a full SUbcategory of an abelian category A, and assume M is closed under , = = extensions in A. Then one can form a new category Q(M) having the same objects as ')0\ , = =, = but :in which a morphism from 101 ' to 10\ is taken to be an isomorphism of MI with a subquotient M,IM of M, where MoC 101, are aubobjects of M such that 101 and MlM, o 0 are objects of ~. Assuming 'the isomorphism classes of objects of ~ form a set, the, cstegory Q(M)= has a classifying space llQ(M)= determined up to homotopy equivalence. One can show that the fundamental group of this classifying spacs is canonically isomor- phic to the Grothendieck group of ~ which motivates dsfining a ssquenoe of X-groups by the formula It is ths goal of the present paper to show that this definition leads to an interesting theory. The first part pf the paper is concerned with the general theory of these X-groups. Section 1 contains various tools for working .~th the classifying specs of a small category. It concludes ~~th an important result which identifies ·the homotopy-theoretic fibre of the map of classifying spaces induced by a.functor. In X-theory this is used to obtain long exsct sequences of X-groups from the exact homotopy sequence of a map.
  • Math 615: Lecture of April 16, 2007 We Next Note the Following Fact

    Math 615: Lecture of April 16, 2007 We Next Note the Following Fact

    Math 615: Lecture of April 16, 2007 We next note the following fact: n Proposition. Let R be any ring and F = R a free module. If f1, . , fn ∈ F generate F , then f1, . , fn is a free basis for F . n n Proof. We have a surjection R F that maps ei ∈ R to fi. Call the kernel N. Since F is free, the map splits, and we have Rn =∼ F ⊕ N. Then N is a homomorphic image of Rn, and so is finitely generated. If N 6= 0, we may preserve this while localizing at a suitable maximal ideal m of R. We may therefore assume that (R, m, K) is quasilocal. Now apply n ∼ n K ⊗R . We find that K = K ⊕ N/mN. Thus, N = mN, and so N = 0. The final step in our variant proof of the Hilbert syzygy theorem is the following: Lemma. Let R = K[x1, . , xn] be a polynomial ring over a field K, let F be a free R- module with ordered free basis e1, . , es, and fix any monomial order on F . Let M ⊆ F be such that in(M) is generated by a subset of e1, . , es, i.e., such that M has a Gr¨obner basis whose initial terms are a subset of e1, . , es. Then M and F/M are R-free. Proof. Let S be the subset of e1, . , es generating in(M), and suppose that S has r ∼ s−r elements. Let T = {e1, . , es} − S, which has s − r elements. Let G = R be the free submodule of F spanned by T .
  • Operations on Metric Thickenings

    Operations on Metric Thickenings

    Operations on Metric Thickenings Henry Adams Johnathan Bush Joshua Mirth Colorado State University Colorado, USA [email protected] Many simplicial complexes arising in practice have an associated metric space structure on the vertex set but not on the complex, e.g. the Vietoris–Rips complex in applied topology. We formalize a remedy by introducing a category of simplicial metric thickenings whose objects have a natural realization as metric spaces. The properties of this category allow us to prove that, for a large class of thickenings including Vietoris–Rips and Cechˇ thickenings, the product of metric thickenings is homotopy equivalent to the metric thickenings of product spaces, and similarly for wedge sums. 1 Introduction Applied topology studies geometric complexes such as the Vietoris–Rips and Cechˇ simplicial complexes. These are constructed out of metric spaces by combining nearby points into simplices. We observe that proofs of statements related to the topology of Vietoris–Rips and Cechˇ simplicial complexes often contain a considerable amount of overlap, even between the different conventions within each case (for example, ≤ versus <). We attempt to abstract away from the particularities of these constructions and consider instead a type of simplicial metric thickening object. Along these lines, we give a natural categorical setting for so-called simplicial metric thickenings [3]. In Sections 2 and 3, we provide motivation and briefly summarize related work. Then, in Section 4, we introduce the definition of our main objects of study: the category MetTh of simplicial metric thick- m enings and the associated metric realization functor from MetTh to the category of metric spaces.
  • Van Kampen Colimits As Bicolimits in Span*

    Van Kampen Colimits As Bicolimits in Span*

    Van Kampen colimits as bicolimits in Span? Tobias Heindel1 and Pawe lSoboci´nski2 1 Abt. f¨urInformatik und angewandte kw, Universit¨atDuisburg-Essen, Germany 2 ECS, University of Southampton, United Kingdom Abstract. The exactness properties of coproducts in extensive categories and pushouts along monos in adhesive categories have found various applications in theoretical computer science, e.g. in program semantics, data type theory and rewriting. We show that these properties can be understood as a single universal property in the associated bicategory of spans. To this end, we first provide a general notion of Van Kampen cocone that specialises to the above colimits. The main result states that Van Kampen cocones can be characterised as exactly those diagrams in that induce bicolimit diagrams in the bicategory of spans Span , C C provided that C has pullbacks and enough colimits. Introduction The interplay between limits and colimits is a research topic with several applica- tions in theoretical computer science, including the solution of recursive domain equations, using the coincidence of limits and colimits. Research on this general topic has identified several classes of categories in which limits and colimits relate to each other in useful ways; extensive categories [5] and adhesive categories [21] are two examples of such classes. Extensive categories [5] have coproducts that are “well-behaved” with respect to pullbacks; more concretely, they are disjoint and universal. Extensivity has been used by mathematicians [4] and computer scientists [25] alike. In the presence of products, extensive categories are distributive [5] and thus can be used, for instance, to model circuits [28] or to give models of specifications [11].
  • LOOP SPACES for the Q-CONSTRUCTION Charles H

    LOOP SPACES for the Q-CONSTRUCTION Charles H

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 52 (1988) l-30 North-Holland LOOP SPACES FOR THE Q-CONSTRUCTION Charles H. GIFFEN* Department of Mathematics, University of Virginia, Charlottesville, VA 22903, U.S. A Communicated by C.A. Weibel Received 4 November 1985 The algebraic K-groups of an exact category Ml are defined by Quillen as KJ.4 = TT,+~(Qkd), i 2 0, where QM is a category known as the Q-construction on M. For a ring R, K,P, = K,R (the usual algebraic K-groups of R) where P, is the category of finitely generated projective right R-modules. Previous study of K,R has required not only the Q-construction, but also a model for the loop space of QP,, known as the Y’S-construction. Unfortunately, the S-‘S-construction does not yield a loop space for QfLQ when M is arbitrary. In this paper, two useful models of a loop space for Qu, with no restriction on the exact category L&, are described. Moreover, these constructions are shown to be directly related to the Y’S- construction. The simpler of the two constructions fails to have a certain symmetry property with respect to dualization of the exact category mm. This deficiency is eliminated in the second construction, which is somewhat more complicated. Applications are given to the relative algebraic K-theory of an exact functor of exact categories, with special attention given to the case when the exact functor is cofinal.
  • Metric Geometry, Non-Positive Curvature and Complexes

    Metric Geometry, Non-Positive Curvature and Complexes

    METRIC GEOMETRY, NON-POSITIVE CURVATURE AND COMPLEXES TOM M. W. NYE . These notes give mathematical background on CAT(0) spaces and related geometry. They have been prepared for use by students on the International PhD course in Non- linear Statistics, Copenhagen, June 2017. 1. METRIC GEOMETRY AND THE CAT(0) CONDITION Let X be a set. A metric on X is a map d : X × X ! R≥0 such that (1) d(x; y) = d(y; x) 8x; y 2 X, (2) d(x; y) = 0 , x = y, and (3) d(x; z) ≤ d(x; y) + d(y; z) 8x; y; z 2 X. On its own, a metric does not give enough structure to enable us to do useful statistics on X: for example, consider any set X equipped with the metric d(x; y) = 1 whenever x 6= y. A geodesic path between x; y 2 X is a map γ : [0; `] ⊂ R ! X such that (1) γ(0) = x, γ(`) = y, and (2) d(γ(t); γ(t0)) = jt − t0j 8t; t0 2 [0; `]. We will use the notation Γ(x; y) ⊂ X to be the image of the path γ and call this a geodesic segment (or just geodesic for short). A path γ : [0; `] ⊂ R ! X is locally geodesic if there exists > 0 such that property (2) holds whever jt − t0j < . This is a weaker condition than being a geodesic path. The set X is called a geodesic metric space if there is (at least one) geodesic path be- tween every pair of points in X.
  • Arxiv:2002.10139V1 [Math.AC]

    Arxiv:2002.10139V1 [Math.AC]

    SOME RESULTS ON PURE IDEALS AND TRACE IDEALS OF PROJECTIVE MODULES ABOLFAZL TARIZADEH Abstract. Let R be a commutative ring with the unit element. It is shown that an ideal I in R is pure if and only if Ann(f)+I = R for all f ∈ I. If J is the trace of a projective R-module M, we prove that J is generated by the “coordinates” of M and JM = M. These lead to a few new results and alternative proofs for some known results. 1. Introduction and Preliminaries The concept of the trace ideals of modules has been the subject of research by some mathematicians around late 50’s until late 70’s and has again been active in recent years (see, e.g. [3], [5], [7], [8], [9], [11], [18] and [19]). This paper deals with some results on the trace ideals of projective modules. We begin with a few results on pure ideals which are used in their comparison with trace ideals in the sequel. After a few preliminaries in the present section, in section 2 a new characterization of pure ideals is given (Theorem 2.1) which is followed by some corol- laries. Section 3 is devoted to the trace ideal of projective modules. Theorem 3.1 gives a characterization of the trace ideal of a projective module in terms of the ideal generated by the “coordinates” of the ele- ments of the module. This characterization enables us to deduce some new results on the trace ideal of projective modules like the statement arXiv:2002.10139v2 [math.AC] 13 Jul 2021 on the trace ideal of the tensor product of two modules for which one of them is projective (Corollary 3.6), and some alternative proofs for a few known results such as Corollary 3.5 which shows that the trace ideal of a projective module is a pure ideal.
  • Combinatorial Topology

    Combinatorial Topology

    Chapter 6 Basics of Combinatorial Topology 6.1 Simplicial and Polyhedral Complexes In order to study and manipulate complex shapes it is convenient to discretize these shapes and to view them as the union of simple building blocks glued together in a “clean fashion”. The building blocks should be simple geometric objects, for example, points, lines segments, triangles, tehrahedra and more generally simplices, or even convex polytopes. We will begin by using simplices as building blocks. The material presented in this chapter consists of the most basic notions of combinatorial topology, going back roughly to the 1900-1930 period and it is covered in nearly every algebraic topology book (certainly the “classics”). A classic text (slightly old fashion especially for the notation and terminology) is Alexandrov [1], Volume 1 and another more “modern” source is Munkres [30]. An excellent treatment from the point of view of computational geometry can be found is Boissonnat and Yvinec [8], especially Chapters 7 and 10. Another fascinating book covering a lot of the basics but devoted mostly to three-dimensional topology and geometry is Thurston [41]. Recall that a simplex is just the convex hull of a finite number of affinely independent points. We also need to define faces, the boundary, and the interior of a simplex. Definition 6.1 Let be any normed affine space, say = Em with its usual Euclidean norm. Given any n+1E affinely independentpoints a ,...,aE in , the n-simplex (or simplex) 0 n E σ defined by a0,...,an is the convex hull of the points a0,...,an,thatis,thesetofallconvex combinations λ a + + λ a ,whereλ + + λ =1andλ 0foralli,0 i n.
  • An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron

    An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron

    An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron. Buchstaber, Victor M and Ray, Nigel 2008 MIMS EPrint: 2008.31 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 Contemporary Mathematics An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron Victor M Buchstaber and Nigel Ray 1. An Invitation Motivation. Sometime around the turn of the recent millennium, those of us in Manchester and Moscow who had been collaborating since the mid-1990s began using the term toric topology to describe our widening interests in certain well-behaved actions of the torus. Little did we realise that, within seven years, a significant international conference would be planned with the subject as its theme, and delightful Japanese hospitality at its heart. When first asked to prepare this article, we fantasised about an authorita- tive and comprehensive survey; one that would lead readers carefully through the foothills above which the subject rises, and provide techniques for gaining sufficient height to glimpse its extensive mathematical vistas. All this, and more, would be illuminated by references to the wonderful Osaka lectures! Soon afterwards, however, reality took hold, and we began to appreciate that such a task could not be completed to our satisfaction within the timescale avail- able. Simultaneously, we understood that at least as valuable a service could be rendered to conference participants by an invitation to a wider mathematical au- dience - an invitation to savour the atmosphere and texture of the subject, to consider its geology and history in terms of selected examples and representative literature, to glimpse its exciting future through ongoing projects; and perhaps to locate favourite Osaka lectures within a novel conceptual framework.