TA2006-Cours

Informations

• Publié par: Vewyur
• Nombre de lectures: 46
• Langue: English

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An introduction to algebraic topology
1
Course at Paris VI University, 2005/2006
Pierre Schapira
May 11, 2006
1
To the students: the material covered by these Notes goes beyond the con-
tents of the actual course. All along the semester, the students will be informed
of what is required for the exam.2Contents
1 Linear algebra over a ring 7
1.1 Modules and linear maps . . . . . . . . . . . . . . . . . . . . . 7
1.2 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 The functor Hom . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6 Koszul complexes . . . . . . . . . . . . . . . . . . . . . . . . . 31
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 The language of categories 37
2.1 Categories and functors . . . . . . . . . . . . . . . . . . . . . . 37
2.2 The Yoneda Lemma . . . . . . . . . . . . . . . . . . . . . . . 42
2.3 Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 Exact functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.7 Filtrant inductive limits . . . . . . . . . . . . . . . . . . . . . 53
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3 Additive categories 59
3.1 Additive . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Complexes in additive categories . . . . . . . . . . . . . . . . . 61
3.3 Simplicial constructions . . . . . . . . . . . . . . . . . . . . . 64
3.4 Double complexes . . . . . . . . . . . . . . . . . . . . . . . . . 66
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 Abelian categories 71
4.1 Abelian . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Complexes in abelian categories . . . . . . . . . . . . . . . . . 75
4.3 Application to Koszul complexes . . . . . . . . . . . . . . . . 79
4.4 Injective objects . . . . . . . . . . . . . . . . . . . . . . . . . . 81
34 CONTENTS
4.5 Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.6 Derived functors . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.7 Bifunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Abelian sheaves 95
5.1 Presheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3 Sheaf associated with a presheaf . . . . . . . . . . . . . . . . . 100
5.4 Internal operations . . . . . . . . . . . . . . . . . . . . . . . . 104
5.5 Direct and inverse images . . . . . . . . . . . . . . . . . . . . 107
5.6 Sheaves associated with a locally closed subset . . . . . . . . . 112
5.7 Locally constant and locally free sheaves . . . . . . . . . . . . 115
5.8 Gluing sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6 Cohomology of sheaves 123
6.1 of sheaves . . . . . . . . . . . . . . . . . . . . . . 123
6.2 Cech complexes for closed coverings . . . . . . . . . . . . . . . 126
6.3 Invariance by homotopy . . . . . . . . . . . . . . . . . . . . . 127
6.4 Cohomology of some classical manifolds . . . . . . . . . . . . . 133
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7 Homotopy and fundamental groupoid 139
7.1 Fundamental groupoid . . . . . . . . . . . . . . . . . . . . . . 139
7.2 Monodromy of locally constant sheaves . . . . . . . . . . . . . 142
7.3 The Van Kampen theorem . . . . . . . . . . . . . . . . . . . . 146
7.4 Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152CONTENTS 5
Introduction
This course is an rst introduction to Algebraic Topology from the point of
view of Sheaf Theory. An expanded version of these Notes may be found in
[23], [24].
Algebraic Topology is usually approached via the study of of homology
de ned using chain complexes and the fundamental group, whereas, here,
the accent is put on the language of categories and sheaves, with particular
attention to locally constant sheaves.
Sheaves on topological spaces were invented by Jean Leray as a tool to
deduce global properties from local ones. This tool turned out to be ex-
tremely powerful, and applies to many areas of Mathematics, from Algebraic
Geometry to Quantum Field Theory.
The functor associating to a sheaf F on a topological space X the space
F(X) of its global sections is left exact, but not right exact in general. The
jderived functors H (X;F) encode the ‘ ‘obstructions" to pass from local to
jglobal. Given a ring k, the cohomology groups H (X;k ) of the sheaf kX X
of k-valued locally constant functions is therefore a topological invariant of
the space X. Indeed, it is a homotopy invariant, and we shall explain how
jto calculate H (X;k ) in various situations.X
We also introduce the fundamental group (X) of a topological space1
(with suitable assumptions on the space) and prove an equivalence of cate-
gories between that of nite dimensional representations of this group and
that of local systems on X. As a byproduct, we deduce the Van Kampen
theorem from the theorem on the glueing of sheaves de ned on a covering.
Lectures will be organized as follows.
Chapter 1 is a brief survey of linear algebra over a ring. It serves as a
guide for the theory of additive and abelian categories which is exposed in
the subsequent chapters.
In Chapter 2 we expose the basic language of categories and functors.
A key point is the Yoneda lemma, which asserts that a categoryC may be
^embedded in the categoryC of contravariant functors onC with values in
the category Set of sets. This naturally leads to the concept of representable
functor. Next, we study inductive and projective limits in some detail and
with many examples.
Chapters 3 and 4 are devoted to additive and abelian categories. The
aim is the construction and the study of the derived functors of a left (or
right) exact functor F of abelian categories. Hence, we start by studying
complexes (and double complexes) in additive and abelian categories. Then
we brie y explain the construction of the right derived functor by using
injective resolutions and later, by using F-injective resolutions. We apply6 CONTENTS
these results to the case of the functors Ext and Tor.
In Chapter 5, we study abelian sheaves on topological spaces (with a
brief look at Grothendieck topologies). We construct the sheaf associated
with a presheaf and the usual internal operations (Hom and
) and external
operations (direct and inverse images). We also explain how to obtain locally
constant or locally free sheaves when glueing sheaves.
In Chapter 6 we prove that the category of abelian sheaves has enough
injectives and we de ne the cohomology of sheaves. We construct resolutions
of sheaves using open or closed Cech coverings and, using the fact that the
cohomology of locally constant sheaves is a homotopy invariant, we show how
to compute the cohomology of spaces by using cellular decomposition. We
apply this technique to deduce the cohomology of some classical manifolds.
In Chapter 7, we de ne the fundamental groupoid (X) of a locally1
arcwise connected space X as well as the monodromy of a locally constant
sheaf and prove that under suitable assumptions, the monodromy functor is
an equivalence. We also show that the Van Kampen theorem may be deduced
from the theorem on the glueing of sheaves and apply it in some particular
sitations.
Conventions. In these Notes, all rings are unital and associative but not
necessarily commutative. The operations, the zero element, and the unit are
denoted by +;; 0; 1, respectively. However, we shall often write for short ab
instead of ab.
All along these Notes, k will denote a commutative ring. (Sometimes, k
will be a eld.)
We denote by; the empty set and byfptg a set with one element.
We by N the set of non-negative integers, N =f0; 1;:::g.Chapter 1
Linear algebra over a ring
This chapter is a short review of basic and classical notions of commutative
algebra.
Many notions introduced in this chapter will be repeated later in a more
general setting.
Some references: [1], [4].
1.1 Modules and linear maps
All along these Notes, k is a commutative ring.
Let A be a k-algebra, that is, a ring endowed with a morphism of rings
’: k! A such that the image of k is contained in the center of A. Notice
that a ring A is always a Z-algebra. If A is commutative, then A is an
A-algebra.
Since we do not assumeA is commutative, we have to distinguish between
left and right structures. Unless otherwise speci ed, a module M over A
means a left A-module.
Recall that an A-module M is an additive group (whose operations and
zero element are denoted +; 0) endowed with an external law AM!M
satisfying: 8> (ab)m =a(bm)<
(a +b)m =am +bm
0 0a(m +m ) =am +am>:
1m =m
0where a;b2A