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Equivariant geometric K-homology with coefficients

Michael Walter

Equivariant geometric K-homology with coefficients

Diplomarbeit

vorgelegt von

Michael Walter

geboren in

Lahr

angefertigt am Mathematischen Institut der Georg-August-Universität zu Göttingen

2010

v

Equivariant geometric K-homology with coefficients

Michael Walter

Abstract

K-homology is the dual of K-theory. Kasparov’s analytic version, where cycles are given by (abstract) elliptic operators over (not necessarily commutative) spaces, has proved to be an extremely powerful tool which, together with its bivariant generalization KK-theory, lies at the heart of many important results at the intersection of algebraic topology, functional analysis and geometry.

Independently, Baum and Douglas have proposed a geometric version of K-homology inspired by singular bordism. Cycles for this theory are given by vector bundles over compact Spinc- manifolds with boundary which map to the target, i.e.

E

f

(M, M)

(X, Y).
There is a natural transformation to analytic K-homology defined by sending such a cycle to the pushforward of the class determined by the twisted Dirac operator. It is well-known to be an isomorphism, although a rigorous proof has appeared only recently.

While both theories have obvious generalizations to the equivariant case and coefficients, the question whether these remain isomorphic is far from trivial (and has negative answer in the general case). In their work on equivariant correspondences Emerson and Meyer have isolated a useful sufficient condition for their theory which, while vastly more general, only deals with the absolute case. Our focus is not so much to construct a geometric theory in the most general situation, but to show that in the presence of a group action and coefficients the above picture still gives a generalized homology theory in a very geometrical way, isomorphic to Kasparov’s theory.

In this thesis we will show that equivariant geometric and analytic K-homology for compact Lie group actions and unital coefficient algebras are naturally isomorphic generalized homology theories on a broad category of spaces, following an approach by Baum, Oyono-Oyono and Schick.

Chapter 1 gives a modern overview of both analytic K-theory and K-homology in terms of Kasparov’s equivariant KK-theory. On the way we generalize several well-known results and constructions to the equivariant case and coefficients, in particular Swan’s theorem, Poincaré duality, Thom isomorphisms and functorial Gysin maps. For the latter we work out in some detail the relationship between its geometrical and analytical descriptions.

Chapter 2 introduces geometric K-homology groups by defining cycles and equivalence relations. By translating the latter into analytical terms we show that the natural transformation from geometric to analytic cycles is well-defined on the level of homology classes. Using the Mostow embedding theorem we then prove that this map is in fact an isomorphism.

vi

Acknowledgements. It is a pleasure to acknowledge Prof. Thomas Schick for inspiring discussions and helpful advice, and for sharing his enthusiasm for doing mathematics.

Contents

  • Abstract
  • v

Chapter 1. Analytic K-theory and K-homology
1.1. C -algebras and topological spaces with group actions 1.2. Hilbert G-C -modules
115

  • 1.3. Equivariant KK-theory
  • 9

1.4. Analytic K-theory 1.5. Topological K-theory 1.6. Analytic K-homology 1.7. Products
15 18 25 25 29 34 40 51 60 62 65 70
1.8. Elliptic operators and analytic K-homology 1.9. Dirac operators 1.10. Spinc-structure and spinor bundles 1.11. Fundamental classes and Poincaré duality 1.12. The double of a manifold 1.13. Equivariant Spinc-structure of the spheres 1.14. Bott periodicity and Thom isomorphism 1.15. Gysin maps

  • Chapter 2. Geometric K-homology
  • 75

75 79 82
2.1. Cycles and equivalence relations 2.2. Geometric K-homology groups 2.3. Isomorphism between geometric and analytic K-homology

  • Outlook
  • 87

89 91 95
Bibliography Notation Index Subject Index

vii

CHAPTER 1

Analytic K-theory and K-homology

In his landmark article [Kas81], Kasparov introduced a bivariant theory, KK-theory, which unified both K-theory and K-homology. Its main feature is the existence of the Kasparov product, an associative pairing

  • KK(A, B) KK(B, C)
  • KK(A, C)

which unifies and generalizes most constructions of the preceding theories. Pushforwards and pullbacks, boundary maps and the pairing of K-theory and K-homology, as well also more involved constructions such as Bott periodicity and the Thom isomorphism can all be expressed in terms of Kasparov products with suitable elements. In the subsequent paper [Kas88], KK-theory was extended to the equivariant setting.

In this chapter, we start by giving a modern account of analytic K-theory and K-homology which we define in terms of Kasparov’s equivariant KK-bifunctor (Sections 1.1–1.7). We shall also generalize the classical topological picture of K-theory to the equivariant case and coefficients (Section 1.5). Key ingredient to this latter result is a version of Swan’s theorem for equivariant bundles of finitely generated, projective Hilbert modules.

The second part of the chapter focuses on connecting analysis with geometry. We recall how symmetric elliptic operators, in particular Dirac operators, define cycles in K-homology and show that it is possible to include into these the action of more general algebras which are similarly well-behaved as C0(M) (Section 1.8). Kasparov’s Dirac element fits naturally into this setting, as do the fundamental classes determined by manifolds equipped with an equivariant Spinc-structure, the K-theory equivalent of an orientation (Sections 1.9–1.11). We will in fact see that both elements are but different sides of the same medal. This will be our way of deducing Poincaré duality. With this machinery in place, we proceed to re-derive Bott periodicity and the Thom isomorphism by a careful analysis of the Spinc-structure on even-dimensional spheres (Sections 1.13 and 1.14). Our proof will give both a topological and an analytic description of these results which will be crucial for the second part of the thesis. In the last section we similarly define “wrong-way” functorial Gysin maps first in an analytical way and then derive concrete geometric descriptions for them (Section 1.15).

Although Kasparov’s theory can be developed in a more general context, it is sufficient for our purposes to only consider separable complex C -algebras, and we can avoid much technical trouble that way. See [Bla98, Chapter 14] for a convenient review of the basic theory of C -algebras and [Bla06] for a more detailed account. We shall also require that all topological spaces be locally

compact, second-countable and Hausdorff and that topological groups even be compact.

1.1. C -algebras and topological spaces with group actions

In this section we will recall the fundamental notion of a G-C -algebra, that is, a C -algebra equipped with a continuous group action by automorphisms. We shall always identify a group element with the automorphism by which it acts.

1.1.1 DEFINITION (G-CONTINUITY, G-C -ALGEBRA). Let A be a C -algebra equipped with a (left)

  • action of a (compact topological) group G by automorphisms. An element a
  • A is called G-

  • continuous if the orbit map g
  • g(a) is continuous.

1

  • 2
  • 1. ANALYTIC K-THEORY AND K-HOMOLOGY

The category of G-C -algebras (G-algebra in [Kas88], covariant system in [Bla98]) is then defined as follows. Its objects are (separable) C -algebras with an action of G by automorphisms such that every element is G-continuous. In other words, the associated group homomorphism G Aut(A) is strongly continuous. Morphisms are G-equivariant -homomorphisms.

The category of (Z/2-)graded G-C -algebras is defined similarly except that we require both the group actions and morphisms to preserve the grading. Unless noted otherwise, all G-C -algebras are assumed to be ungraded (or, equivalently, trivially graded). We will always denote the degree of homogeneous elements a by a.

We will always consider the complex numbers C as a trivial G-C -algebra. Recall that the tensor product A B of graded G-C -algebras A and B can be formed in different ways by completing the skew-commutative algebraic tensor product A B. We will always use the minimal/spatial tensor product from [Kas81, Section 2]. There exist canonical braiding iso-

  • morphisms A
  • B
  • B

A sending a tensor a b to ( 1) a bb a such that the class of graded
G-C -algebras forms a symmetric monoidal category. We write A B if both algebras are trivially graded. A detailed treatment of tensor products of C -algebras can be found in [WO93, Appendix T]; for the graded case see [Bla98, Section 14].

Most of the time, one of the factors in our tensor products will be a nuclear C -algebra so that all C -tensor products are in fact equal. We will make repeated use of the following facts:

1.1.2 PROPOSITION ([Bla06, Theorem 15.8.2]). The following classes of C -algebras are nuclear: Commutative C -algebras, finite-dimensional C -algebras, C -algebras of compact operators on a separable Hilbert space.

Moreover, the class of nuclear C -algebras is closed under direct limits, extensions, ideals and quotients.

1.1.3 PROPOSITION ([WO93, Theorem T.6.26]). Let

  • 0
  • 0

  • A
  • A
  • A

be a short exact sequence of C -algebras and B another C -algebra. If at least one of the C -algebras A or B is nuclear then the tensored sequence

  • A
  • B

  • 0
  • 0

  • A
  • B
  • A
  • B

remains exact.

1.1.4 EXAMPLE ([WO93, Corollary T.6.17, Theorem T.6.20]). In particular, the C -algebra C0(X) of complex-valued functions vanishing at infinity on a locally compact Hausdorff space X is nuclear. Let us also recall that there exist canonical isomorphisms

C0(X)

A

C0(X, A),

  • f
  • a
  • (x
  • f (x)a),

  • f (x)g(y))
  • C0(X) C0(Y)

  • C0(X Y),
  • f
  • g

((x, y)

for all graded C -algebras A and spaces Y. Here and in the following, the space C0(X, A) of A-valued functions vanishing at infinity is always equipped with the supremum norm.

We will now formalize our notion of topological spaces and pairs with a group action. 1.1.5 DEFINITION (G-SPACE, G-PAIR). A G-space is a (locally compact, second-countable, Hausdorff) space X with a (left) action of a group G such that the associated map G

X

X is continuous. Morphisms will always be proper G-maps, i.e. proper G-equivariant continuous maps.

If Y

(X, Y)

X is a closed G-invariant subspace then (X, Y) is called a G-pair. A morphism of G-pairs
(X , Y ) is a proper G-map X X sending Y to Y . We define the Cartesian product of G-pairs (X, Y) and (X , Y ) to be the pair (X X , X

  • Y
  • Y

X ). Similarly, we define their

disjoint union to be the pair (X X , Y Y ).

  • 1.1. C -ALGEBRAS AND TOPOLOGICAL SPACES WITH GROUP ACTIONS
  • 3

1.1.6 LEMMA. The group action of a non-compact G-space X extends naturally to its one-point compact-

  • ification X+ = X
  • (with the trivial action on ).

  • PROOF. It suffices to verify continuity of G X+
  • X+ in the point (1G, ). Let X+ K with

X+. Then GK, the orbit of K, is a compact set in X and
(X+ GK) is a neighborhood of (1G, ) that maps into X+ K.

KG

X compact be a neighborhood of

ꢀ

In the following, we will often consider bifunctors defined for G-pairs together with graded G- C -algebras, i.e. functors whose domain is the product of the category of G-pairs with the category of graded G-C -algebras (or its opposite). We will write objects in this category in the form (X, Y; A) and morphisms as (ϕ; Φ) where ϕ is a morphism of G-pairs and Φ a morphism of graded G-C -algebras. We will usually omit the G-C -algebras if A = C and the subspace Y if it is empty.

1.1.7 PROPOSITION. The assignment

(X, Y; A)

  • C0(X Y, A) C0(X Y)
  • A

ꢀꢀ

C0 :

(ϕ; Φ)

(f

Φ

f

ϕ

)

X Y

defines a bifunctor which is contravariant in the G-pair and covariant in the algebra variable. Here, we extend functions by zero so that the second composition makes sense, and G acts on C0(X Y, A) by the formula g(f )(x) := g(f (g 1(x))).

PROOF. This is obvious once we have established that the functor is well-defined on objects and morphisms.

Let (X, Y) be a G-pair and A a graded G-C -algebra. It is well-known that C0(X Y, A) is a C - algebra and that C0(X Y, A) C0(X Y) A in the sense of graded C -algebras (Example 1.1.4). It is separable because the space X is assumed to be second-countable. Since the G-action defined by the above formula corresponds precisely to the diagonal action on the tensor product it suffices to show that C0(X Y) is in fact a G-C -algebra, i.e. that every element is G-continuous. Clearly we may assume that Y = . Then the orbit map of any function f

C0(X)

C(X+) can be written as the composition

1

  • g
  • f

C(X+, X+)

C(X+)

G

which in fact maps into C0(X). Here, the function spaces C(X+, X+) and C(X+) are equipped with the compact-open topology so that both maps are continuous. But the compact-open topology on the latter space agrees with the topology induced by the supremum norm. This shows that f is G-continuous.

It remains to verify that C0 is well-defined on morphisms. We only need to show that the preimage of a compact set K X Y under ϕ X Y is compact (we do not have to worry about Φ because it is norm-decreasing). But this preimage is precisely ϕ 1(K) (X Y) = ϕ 1(K), hence compact in X (since ϕ is proper), and thus compact in the open subset X Y containing it.

ꢀ

1.1.8 REMARK. The well-known Gelfand-Naimark theorem asserts that this functor implements a duality between the category of spaces and the category of commutative C -algebras (see e.g. [Bla06, Theorem II.2.2.4]).

1.1.9 REMARK. The functor C0 maps disjoint unions to direct sums and Cartesian products to tensor products. Indeed,

C0((X X ) (Y Y )) = C0((X Y) (X Y )) = C0(X Y) C0(X Y ),

  • C0((X X ) (X
  • Y
  • Y
  • X )) = C0((X Y) (X Y )) = C0(X Y) C0(X Y ).

  • 4
  • 1. ANALYTIC K-THEORY AND K-HOMOLOGY

Let us write ϕ := C0(ϕ; idA) and Φ := C0(id(X,Y); Φ) for the pullback of pairs and the pushforward of algebras, respectively. There are two natural morphisms associated to every G-pair

  • (X, Y), the inclusion incl: Y X of the closed subspace and the relativization map rel: X
  • (X, Y).

1.1.10 COROLLARY. We have natural short exact sequences of G-C -algebras

  • rel
  • incl

(1.1.11)

  • C0(X Y, A)
  • C0(X, A)
  • C0(Y, A)

  • 0
  • 0

which are semisplit, i.e. there exists a completely positive, norm-decreasing, grading-preserving C-linear section for the projection incl .

PROOF. It is is easy to see that the above sequence is exact for A = C, and because C0(Y) is nuclear it is also semisplit [Bla98, Theorem 15.8.3]. Moreover, by nuclearity the sequence remains exact after tensoring with A (Proposition 1.1.3) and it is clear that the resulting sequence is still semisplit (tensor the split with the identity map of A).

  • For naturality, observe that every morphism (ϕ; Φ): (X, Y; A)
  • (X , Y ; B) gives rise to a com-

mutative diagram

  • (Y, ; A)
  • (X, ; A)
  • (X, Y; A)

ꢀꢀ

  • (ϕ;Φ)
  • (ϕ;Φ)
  • (ϕ Y;Φ)

  • (Y , ; B)
  • (X , ; B)

(X , Y ; B)

in the product category. The claim thus follows from applying the functor C0.

ꢀ

These basic results will enable our later passage from topology to analysis. Our last two examples are finite-dimensional graded G-C -algebras. They will play a crucial role in what follows.

1.1.12 EXAMPLE (ENDOMORPHISMS OF THE EXTERIOR ALGEBRA). Let W be a finite-dimensional

unitary representation1 of the group G with isometric antilinear involution x

x

such that the group action commutes with the involution. We denote by W the same representation with involution x x . The inner product and G-action extend to the exterior algebra which then also becomes a finite-dimensional unitary representation of G, graded into the span of even and odd monomials. In particular, the space L( W) of linear endomorphisms of the

W

exterior algebra, graded into grading-preserving and grading-reversing operators, is a graded G- C -algebra — this is easy to see directly, but also follows from Example 1.2.3 in the next section.

1.1.13 EXAMPLE (CLIFFORD ALGEBRA). Let W as in the previous example. The complex Clifford algebra CW is defined as quotient of the tensor algebra of W with respect to the relation x2

=

x , x . It inherits the G-action and Z/2-grading from the tensor algebra. We obtain an antilinear involution on CW by setting (x1 xk) := xk x1 for x1, . . . , xk W.

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    Cambridge University Press 978-1-107-15074-4 — Singular Intersection Homology Greg Friedman Index More Information Index KEY: Numbers in bold indicate where terms are defined ∂-pseudomanifold, see pseudomanifold, effects of simplicial subdivision on, 96–98 ∂-pseudomanifold in small degrees, 130 ∂-stratified pseudomanifold, see stratified on regular strata, 96, 130 pseudomanifold, ∂-stratified pseudomanifold PL chain, 121 abstract simplicial complex, see simplicial complex, simplex, 92 abstract singular chain, 129 acknowledgments, xiv singular simplex, 129 acyclic model, 199 strata vs. skeleta, 105–107 Acyclic Model Theorem, 368 Aluffi, Paolo, 707 admissible triangulation, see triangulation, admissible assembly map, 697, 702 agreeable triple of perversities, 372, 376 AT map, 751 ( , ) AT space, 744, 751 Qp¯ ,t¯Y Qt¯X ,q¯ ; Q , 434 (0¯, t¯; 0¯), 373 augmentation cocycle (1), xxi (n¯, n¯; 0¯), 374 augmentation map (a), xx see (p¯, Dp¯; 0¯), 373 augmented intersection chain complex, (p¯, t¯;¯p), 374 intersection chain complex, augmented (t¯, 0;¯ 0¯), 373 balls, 189 and diagonal maps, 372 Banagl, Markus, xiv, 695, 697, 699, 701, 704, 707, 708 and projection maps, 398 basic sets, 607 in terms of dual perversities, 373 Bernstein, Joseph, 710 Akin, Ethan, 692, 706 Beshears, Aaron, 59 Albin, Pierre, 706, 707, 709 Be˘ılinson, Alexander, 710 Alexander–Whitney map, 199, 365–367 bilinear pairing, see pairing intersection version, see intersection Bockstein map, 233 Alexander–Whitney map (IAW) bordism, 689–702 algebra background, 713–738 bordism group, 690
  • UC Berkeley UC Berkeley Electronic Theses and Dissertations

    UC Berkeley UC Berkeley Electronic Theses and Dissertations

    UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Combinatorial Topology and Applications to Quantum Field Theory Permalink https://escholarship.org/uc/item/7r44w49f Author Thorngren, Ryan George Publication Date 2018 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Combinatorial Topology and Applications to Quantum Field Theory by Ryan George Thorngren A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Vivek Shende, Chair Professor Ian Agol Professor Constantin Teleman Professor Joel Moore Fall 2018 Abstract Combinatorial Topology and Applications to Quantum Field Theory by Ryan George Thorngren Doctor of Philosophy in Mathematics University of California, Berkeley Professor Vivek Shende, Chair Topology has become increasingly important in the study of many-body quantum mechanics, in both high energy and condensed matter applications. While the importance of smooth topology has long been appreciated in this context, especially with the rise of index theory, torsion phenomena and dis- crete group symmetries are relatively new directions. In this thesis, I collect some mathematical results and conjectures that I have encountered in the exploration of these new topics. I also give an introduction to some quantum field theory topics I hope will be accessible to topologists. 1 To my loving parents, kind friends, and patient teachers. i Contents I Discrete Topology Toolbox1 1 Basics4 1.1 Discrete Spaces..........................4 1.1.1 Cellular Maps and Cellular Approximation.......6 1.1.2 Triangulations and Barycentric Subdivision......6 1.1.3 PL-Manifolds and Combinatorial Duality........8 1.1.4 Discrete Morse Flows...................9 1.2 Chains, Cycles, Cochains, Cocycles..............
  • Spanier-Whitehead K-Duality for $ C^* $-Algebras

    Spanier-Whitehead K-Duality for $ C^* $-Algebras

    SPANIER-WHITEHEAD K-DUALITY FOR C∗-ALGEBRAS JEROME KAMINKER AND CLAUDE L. SCHOCHET Abstract. Classical Spanier-Whitehead duality was introduced for the stable homotopy category of finite CW complexes. Here we provide a comprehensive treatment of a noncommutative version, termed Spanier-Whitehead K-duality, which is defined on the category of C∗-algebras whose K-theory is finitely generated and that satisfy the UCT, with morphisms the KK-groups. We explore what happens when these assumptions are relaxed in various ways. In particular, we consider the relationship between Paschke duality and Spanier- Whitehead K-duality. Contents 1. Introduction 1 2. Spanier-Whitehead K- Duality 4 3. Fitting Classical Spanier-Whitehead duality into the Spanier-Whitehead K-duality framework 8 4. Examples of noncommutative duality 10 4.1. Hyperbolic dynamics 10 4.2. Baum-Connes conjecture 11 4.3. Mukai transform 12 5. Poincar´eduality 12 6. Existence of Spanier-Whitehead K-Duals 14 7. Non-existence of Spanier-Whitehead K-Duals 15 8. Mod-p K-theory 16 9. Paschke Duality 17 ∗ 10. C -substitutes I: K∗(A) countable 19 11. C∗-substitutesII:bootstrapentries 22 References 27 arXiv:1609.00409v4 [math.OA] 13 Jun 2017 1. Introduction Classical Spanier-Whitehead duality is a generalization of Alexander duality, which relates the homology of a space to the cohomology of its complement in a sphere. Ed Spanier and J.H.C. Whitehead [42], [43], noting that the dimension of the sphere did not play an essential role, adapted it to the context of stable homo- topy theory. Its history and its relation to other classical duality ideas are described in depth by Becker and Gottlieb [4].
  • Arxiv:1606.04233V2 [Math.AT] 21 Sep 2016 Ls.I 7,M Oek N .Mchro Rv Htthe That Prove Macpherson R

    Arxiv:1606.04233V2 [Math.AT] 21 Sep 2016 Ls.I 7,M Oek N .Mchro Rv Htthe That Prove Macpherson R

    SINGULAR DECOMPOSITIONS OF A CAP PRODUCT DAVID CHATAUR, MARTINTXO SARALEGI-ARANGUREN, AND DANIEL TANRE´ Abstract. In the case of a compact orientable pseudomanifold, a well-known theorem of M. Goresky and R. MacPherson says that the cap product with a fundamental class factorizes through the intersection homology groups. In this work, we show that this classical cap product is compatible with a cap product in intersection (co)- homology, that we have previously introduced. If the pseudomanifold is also normal, for any commutative ring of coefficients, the existence of a classical Poincar´eduality isomorphism is equivalent to the existence of an isomorphism between the intersection homology groups corresponding to the zero and the top perversities. Let X be a compact oriented pseudomanifold and [X] ∈ Hn(X; Z) be its fundamental class. In [7], M. Goresky and R. MacPherson prove that the Poincar´eduality map k defined by the cap product −∩ [X]: H (X; Z) → Hn−k(X; Z) can be factorized as p p k α p β H (X; Z) −→ Hn−k(X; Z) −→ Hn−k(X; Z), (1) p where the groups Hi (X; Z) are the intersection homology groups for the perversity p. The study of the Poincar´eduality map via a filtration on homology classes is also considered in the thesis of C. McCrory [10], [11], using a Zeeman’s spectral sequence. In [5, Section 8.1.6], G. Friedman asks for a factorization of the Poincar´eduality map through a cap product defined in the context of an intersection cohomology recalled in Section 4. He proves it with a restriction on the torsion part of the intersection coho- mology.
  • Introduction to Algebraic Topology and Algebraic

    Introduction to Algebraic Topology and Algebraic

    INTERNATIONAL SCHOOL FOR ADVANCED STUDIES Trieste U. Bruzzo INTRODUCTION TO ALGEBRAIC TOPOLOGY AND ALGEBRAIC GEOMETRY Notes of a course delivered during the academic year 2002/2003 La filosofia `escritta in questo grandissimo libro che continuamente ci sta aperto innanzi a gli occhi (io dico l'universo), ma non si pu`o intendere se prima non si impara a intender la lingua, e conoscer i caratteri, ne' quali `escritto. Egli `escritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure geometriche, senza i quali mezi `eimpossibile a intenderne umanamente parola; senza questi `eun aggirarsi vanamente per un oscuro laberinto. Galileo Galilei (from \Il Saggiatore") i Preface These notes assemble the contents of the introductory courses I have been giving at SISSA since 1995/96. Originally the course was intended as introduction to (complex) algebraic geometry for students with an education in theoretical physics, to help them to master the basic algebraic geometric tools necessary for doing research in algebraically integrable systems and in the geometry of quantum field theory and string theory. This motivation still transpires from the chapters in the second part of these notes. The first part on the contrary is a brief but rather systematic introduction to two topics, singular homology (Chapter 2) and sheaf theory, including their cohomology (Chapter 3). Chapter 1 assembles some basics fact in homological algebra and develops the first rudiments of de Rham cohomology, with the aim of providing an example to the various abstract constructions. Chapter 5 is an introduction to spectral sequences, a rather intricate but very pow- erful computation tool.