TA2006-Cours
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TA2006-Cours

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An introduction to algebraic topology1Course at Paris VI University, 2005/2006Pierre SchapiraMay 11, 20061To 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 informedof what is required for the exam.2Contents1 Linear algebra over a ring 71.1 Modules and linear maps . . . . . . . . . . . . . . . . . . . . . 71.2 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 The functor Hom . . . . . . . . . . . . . . . . . . . . . . . . . 131.4 Tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . 201.5 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.6 Koszul complexes . . . . . . . . . . . . . . . . . . . . . . . . . 31Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 The language of categories 372.1 Categories and functors . . . . . . . . . . . . . . . . . . . . . . 372.2 The Yoneda Lemma . . . . . . . . . . . . . . . . . . . . . . . 422.3 Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . 442.4 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.6 Exact functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.7 Filtrant inductive limits . . . . . . . . . . . . . . . . . . . . . 53Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

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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 and m;m 2M.
Note that M inherits a structure of a k-module via ’. In the sequel, if
there is no risk of confusion, we shall not write ’.
78 CHAPTER 1. LINEAR ALGEBRA OVER A RING
opWe denote by A the ring A with the opposite structure. Hence the
op opproduct ab in A is the product ba in A and an A -module is a right A-
module.
Note that if the ring A is a eld (here, a eld is always commutative),
then an A-module is nothing but a vector space.
Examples 1.1.1. (i) The rst example of a ring is Z, the ring of integers.
Since a eld is a ring, Q;R;C are rings. If A is a commutative ring, then
A[x ;:::;x ], the ring of polynomials inn variables with coe cien ts inA, is1 n
also a commutative ring. It is a sub-ring ofA[[x ;:::;x ]], the ring of formal1 n
powers series with coe cien ts in A.
(ii) Let k be a eld. Then for n > 1, the ring M (k) of square matrices ofn
rank n with entries in k is non commutative.
(iii) Letk be a eld. The Weyl algebra in n variables, denoted W (k), is then
non commutative ring of polynomials in the variables x , @ (1 i;j n)i j
with coe cien ts in k, and relations :
i[x;x ] = 0; [@;@ ] = 0; [@ ;x ] =i j i j j i j
iwhere [p;q] =pq qp and is the Kronecker symbol.j
The Weyl algebra W (k) may be regarded as the ring of di eren tial op-n
erators with coe cien ts in k[x ;:::;x ], and k[x ;:::;x ] becomes a left1 n 1 n
W (k)-module: x acts by multiplication and @ is the derivation with re-n i i
spect to x .i
A morphismf : M!N ofA-modules is anA-linear map, i.e.,f satis es:
0 0 0f(m +m ) =f(m) +f(m ) m;m 2M
f(am) =af(m) m2M;a2A:
A morphism f is an isomorphism if there exists a morphism g :N!M
with fg = id ;gf = id .N M
1If f is bijective, it is easily checked that the inverse map f : N! M
is itself A-linear. Hence f is an isomorphism if and only if f is A-linear and
bijective.
0A submodule N of M is a subset N of M such that n;n 2 N implies
0n + n 2 N and n 2 N;a 2 A implies an 2 N. A submodule of the
A-module A is called an ideal of A. Note that if A is a eld, it has no
non trivial ideal, i.e., its only ideals aref0g and A. If A = C[x], then
I =fP2 C[x];P(0) = 0g is a non trivial ideal.
If N is a submodule of M, the quotient module M=N is characterized
by the following \universal property": for any module L, any morphism;

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1.1. MODULES AND LINEAR MAPS 9
h: M!L which induces 0 on N factorizes uniquely through M=N. This is
visualized by the diagram
g
M=NN M@ @@ @
0@ h@@ 00@ h
L
Let I be a set, and letfMg be a family of A-modules indexed by I. Thei i2IQ
product M is the set of familiesf(x ) g with x 2 M , and this seti i i2I i ii
naturally inherits a structure of an A-module. There are natural surjective
morphisms: Y
: M !M :k i k
i
Note that given a module L and a family of morphisms f : L! M , thisi iQ
family factorizes uniquely through M . This is visualized by the diagramii
Mk ikkk vk vk vf ki k vkk vk vkk v k ik vk vk Qkkk
S ML S kS kSS GS GSS G S jGS GSS GSS GS GSfj S GSSS
M :jL Q
The direct sum M is the submodule of M consisting of familiesi ii i
f(x ) g with x = 0 for all but a nite number of i2 I. In particular,i i2I i L Q
if the setI is nite, the natural injection M ! M is an isomorphism.i ii i
There are natural injective morphisms: M
" :M ! M:k k i
i L
We shall sometimes identifyM to its image in M by" . Note that givenk i ki
a module L and a family of morphisms f : M ! L, this family factorizesi iL
uniquely through M . This is visualized by the diagramii
M Si SH SSH SH SSH S fiSH SH SSH SS" H Si H SSSL SS
Mk L:kk kkkkw kw" kj w kkw kw kkw kkw k fk jw kw kkk
Mj10 CHAPTER 1. LINEAR ALGEBRA OVER A RING
If M =M for all i2I, one writes:i M Y
(I) IM := M; M := M:i i
i i
AnA-moduleM is free of rank one if it is isomorphic toA, andM is freeL
if it is isomorphic to a direct sum L , each L being free of rank one.i ii2I
If card (I) is nite, say r, then r is uniquely determined and one says M is
free of rank r.
Let f : M!N be a morphism of A-modules. One sets :
Kerf = fm2M; f(m) = 0g
Imf = fn2N; there exists m2M; f(m) =ng:
These are submodules of M and N respectively, called the kernel and the
image of f, respectively. One also introduces the cokernel and the coimage
of f:
Coker f =N= Im f; Coimf =M= Kerf:
Since the natural morphism Coimf! Imf is an isomorphism, one shall not
use Coim when dealing with A-modules.
If (M ) is a family of submodules of an A-module M, one denotes byi i2IP
M the submodule of M obtained as the image of the natural morphismiiL S
M !M. This is also the module generated inM by the set M . Onei ii i
calls this module the sum of the M ’s in M.i
Example 1.1.2. Let W (k) denote as above the Weyl algebra. Considern
the left W (k)-linear map W (k)! k[x ;:::;x ], W (k)3 P 7! P(1)2n n 1 n n
k[x ;:::;x ]. This map is clearly surjective and its kernel is the left ideal1 n
generated by (@ ;;@ ). Hence, one has the isomorphism of left W (k)-1 n n
modules: X (1.1) W (k)= W (k)@ !k[x ;:::;x ]:n n j 1 n
j
1.2 Complexes
De nition 1.2.1. A complex M of A-modules is a sequence of modules
j j j 1j j j+1M ;j2 Z and A-linear maps d :M !M such that d d = 0 forM M M
all j.