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Transcript
SECTION 7: PRESHEAVES AND ČECH COHOMOLOGY
Introductory remarks (to be added).
Definition 1. A presheaf on a topological space X is an assignment A of an abelian group A(U )
to every open set U in X, together with group homomorphisms rU V : A(V ) → A(U ) whenever
U ⊆ V satisfying the following conditions:
rU U = id,
rU V ◦ rV W = rU W ,
whenever
U ⊆ V ⊆ W.
Remark 2.
(a) These group homomorphisms are called restriction maps; accordingly, one
often writes
rU V (a) = a|U (for a ∈ A(V )).
The elements of A(U ) are called sections of A over U .
(b) What we have just defined is the notion of a presheaf of abelian groups. In exactly the same
way, one can define the notion of presheaf of R-modules, or for that matter, the notion of
presheaf with values in any given category. In fact, another way to phrase the definition of
a presheaf is as a functor from a suitable category to the category of abelian groups.
(c) As you might expect, a presheaf is something related to a sheaf. We will not discuss sheaf
theory in this course, but a glimpse can be gotten from the exercises.
Definition 3. Let A and B be presheaves. A morphism ϕ : A → B of presheaves consists of a
family of group homomorphisms ϕU : A(U ) → B(U ) compatible with the restriction operations:
for U ⊆ V, a ∈ A(V ).
ϕV (a)|U = ϕU (a|U ),
Example 4.
(a) Let A be an abelian group. Then A defines a constant presheaf, also
denoted A, by A(U ) = A for any open set U , with all restriction maps being identities.
(b) For an open set U ⊆ X, write R(U ) for the set of continuous real-valued functions on U .
e this defines a presheaf R. In a similar way, one defines
With the usual restriction maps,
a presheaf C of complex-valued functions, a presheaf S 1 eof circle-valued functions, and a
e
presheaf A efor any topological (or Lie) group A.
p
p e
(c) Let C (U, A) (or CSing (U, A) for emphasis) be the group of singular cochains with values
in A, as defined in lecture 3. With the obvious restriction maps, these form a presheaf
p
CSing
(−, A).
The differential (coboundary) is a morphism of presheaves
p
p+1
d : CSing
(−, A) → CSing
(−, A).
In fact, what all this amounts to is that we have one big presheaf with values in the category
of chochain complexes. The associated cohomology groups form another presheaf
U 7→ H p (U, A).
1
2
SECTION 7: PRESHEAVES AND ČECH COHOMOLOGY
(d) Let X be a smooth manifold. Then there are similar presheaves of smooth functions with
values in any abelian Lie group A. Another very important presheaf is the presheaf Ωp of
differential p-forms,
Ωp (U ) = {ω | ω is a p-form on U }.
These Ωp for various p together with the de Rham differential again define a presheaf of
cochain complexes.
Remark 5.
(a) Let ϕ : A → B be a morphism of presheaves on X. If for any open U ⊆ X one
takes the kernel
ker(ϕU ) ⊆ A(U ),
then these groups together form another presheaf on X, denoted ker(ϕ), and the evident
inclusion is a morphism of presheaves ker(ϕ) A. In exactly the same way, by taking
images and cokernels, one can form presheaves im(ϕ) and cok(ϕ).
(b) One says that a sequence of presheaf morphisms
A
ϕ
/B
ψ
/C
is exact (at B) if ker(ψ) = im(ϕ). This amounts to the same as saying that for any open
set U ,
/ B(U )
/ C(U )
A(U )
is an exact sequence of abelian groups. Images, kernels and cokernels, and their exactness
properties, behave exactly as for abelian groups. A more formal way of expressing this is
to say that the presheaves on X form an abelian category.
Example 6. Consider the exact sequence
0
/R
/Z
exp
/ S1
/0
of abelian Lie groups (where we view Z as a discrete group). By taking continuous functions with
values in these groups, we obtain a sequence of presheaves
/Z
/ R exp / S 1
/0
e
e
e
on any space X. Notice that the presheaf Z consists of locally constant functions.
e On the other hand, for any simply connected open
This sequence need not be exact. (Why?)
set U , the sequence
/ Z(U )
/ R(U ) exp / S 1 (U )
/0
0
e
e
e
α / 1
is exact. Indeed, a section α ∈ S 1 (U ) is a map U
S , for which we can form the pullback
e
/R
Ũ
0
U
α
/ S1.
Then Ũ → U is a covering space of U . Since U is simply connected, Ũ is simply a sum of copies of
U , and Ũ → U has a section. Such a section then provides a lift U → Ũ → R of α.
SECTION 7: PRESHEAVES AND ČECH COHOMOLOGY
3
Digression 7. There are various (related) ways to repair this typical failure of exactness. One way
is to change the notion of surjectivity and only require “local surjectivity”. For this to be comparable
with the formation of images and cokernels, one has to pass from presheaves to sheaves. Another
is to work with presheaves defined on a basis of open sets only.
Čech cohomology. Let X be a topological space, and let A be a presheaf on X. Let U be
an open cover of X, say U = {Ui }i∈I . It will be convenient (but not necessary, see Exercises) to
assume that I is equipped with a linear order. For a sequence
i0 < i1 < . . . < in
of elemenets of I , we write Ui0 ...in = Ui0 ∩ . . . ∩ Uin and i0 . . . ibk . . . in for the sequence i0 . . . in with
ik left out, so that Ui0 ...in ⊆ Ui0 ...ibk ...in for k = 0, . . . , n. Thus, there are restriction maps which we
all denote by
dk : A(Ui0 ...ibk ...in ) → A(Ui0 ...in ).
One can assemble all these together into a cochain complex as follows: let
Y
C n (U, A) =
A(Ui0 ...in )
i0 <...<in
where the product ranges are sequences with Ui0 ...in 6= ∅, and define
d : C n−1 (U, A) → C n (U, A)
for an element a ∈ C n−1 (U, A), i.e. a sequence a = (ai0 ...in−1 )i0 <...<in−1 , by
(da)i0 ...in =
n
X
(−1)k dk (ai0 ...ibk ...in ).
k=0
We suggest to the reader to think a bit about this somewhat complicated definition, and why it
makes sense.
Lemma 8. This structure (C ∗ (U, A), d) is a cochain complex.
Proof. We have to check that “dd = 0”. So consider two consecutive d’s,
C n−1 (U, A)
d
/ C n (U, A)
d
/ C n+1 (U, A),
and an element a ∈ C n−1 (U, A). Then the coordinate of dda at i0 < . . . < in+1 is
(dda)i0 ...in+1 =
n+1
X
(−1)k (da)i0 ...ibk ...in+1
k=0
while for each term on the right we have
(da)i0 ...ibk ...in+1 =
k−1
X
n+1
X
l=0
l=k+1
(−1)l ai0 ...ibl ...ibk ...in+1 +
(−1)l−1 ai0 ...ibk ...ibl ...in+1 .
Thus
(dda)i0 ...in+1 =
n+1
XX
(−1)k+l ai0 ...ibl ...ibk ...in+1 +
k=0 l<k
and it is clear that these two terms cancel each other.
n+1
XX
(−1)k+l−1 ai0 ...ibk ...ibl ...in+1
k=0 l>k
4
SECTION 7: PRESHEAVES AND ČECH COHOMOLOGY
Definition 9. Let X be a topological space, U an open cover of X, and A a presheaf on X. The
cohomology of the complex (C ∗ (U, A), d) just described is called the Čech cohomology of X with
respect to the cover U, with coefficients in A, and denoted H ∗ (U, A).
(One also simply says Čech cohomology of U, or of X thus leaving the cover U implicit. Recall
that the definition depends on the chosen order on the index set of the cover U. It can be shown,
however, that the resulting cohomology groups do not, see Exercises.)
This definition of the Čech cohomology groups may strike you as somewhat involved, but these
groups are often very easy to calculate if U is small and if A is not complicated, and the calculations
work in much the same way as cellular homology.
Čech cohomology shares several fundamental properties with other cohomology theories. One of
these is the following:
Proposition 10. Let 0 → A → B → D → 0 be a short exact sequence of presheaves on a space X.
Then for any open cover U there is an induced long exact sequence in Čech cohomology,
. . . → H n (U, A) → H n (U, B) → H n (U, D) → H n+1 (U, A) → . . .
Proof. This immediately follows from the short exact sequence of cochain complexes
0 → C ∗ (U, A) → C ∗ (U, B) → C ∗ (U, D) → 0
and the general facts from Algebraic Topology I.
Proposition 11. Let X be a paracompact Hausdorff space and let U be an open cover of X such
that all finite intersections Ui0 ...in are simply connected. Then there is a canonical isomorphism
1
H 1 (U, S 1 ) → H 2 (U, Z).
e
e
(We will see in the next section that H 2 (U, Z) is often isomorphic to the usual singular cohomology
e
H 2 (X, Z).)
Proof. The short exact sequence 0 → Z → S 1 → R → 0 of Example 6 induces a long exact sequence
e
e
e
. . . → H 1 (U, R) → H 1 (U, S 1 ) → H 2 (U, Z) → H 2 (U, R) → . . .
e
e
e
e
It thus suffices to prove the following lemma.
Lemma 12. For an open cover U of paracompact Hausdorff space X, one has
H i (U, R) = 0 for i > 0.
e
Proof. We will only prove this for i = 1, 2 as needed for the proposition. The general case is proved
in exactly the same way, and left as an exercise. The proof uses a partition of unity subordinated
to U, i.e. a family of functions
θi : Ui → [0, 1],
one for each Ui , such that the supports
X of the θi are contained in Ui and form a locally finite
family of closed sets, and such that
θi (x) = 1 for each x ∈ X. We will also use the following
i
conventions: if a ∈ C n (U, R) is a cochain and i0 , . . . , in is a sequence of indices which is not strictly
ordered, then ai0 ...in := 0 eif two of the indices coincide; and otherwise ai0 ...in = (−1)|σ| aσ(i0 )...σ(in )
1Recall that a topological space X is paracompact if every open cover has an open refinement that is locally finite.
A cover is locally finite if each point of the space has a neighbourhood that intersects only finitely many elements of
that cover.
SECTION 7: PRESHEAVES AND ČECH COHOMOLOGY
5
where σ is the permutation putting the indices in the right order, i.e. σ(i0 ) < . . . < σ(in ), and |σ|
is the sign of σ.
To prove that H 1 (U, R) = 0, let a = {aij }i<j be a 1-cocycle. Thus, for any three indices
e
i < j < k,
ajk − aik + aij = 0.
(1)
Now define b = {bi } by
bi =
X
θk aki : Ui → R.
k
(So, by convention, this is
X
θk aki −
k<i
X
θk aik .) We claim that db = a, so that the cohomology
k>i
class [a] vanishes. Indeed, for i < j
(db)ij =
X
θk (akj − aki ).
k
Now look at the three cases k < i, i < k < j and j < k. For k < i we have aij − akj + aki = 0 since
a is a cocycle, and hence
akj − aki = aij .
Similarly, if i < j < k we have akj − aij + aik = 0 by the cocycle condition, and
akj − aki = akj + aik = aij (by our convention).
And if i < j < k, one checks similarly that akj − aki = aij . Thus
X
(db)ij =
θk aij = aij .
k
This proves that H ( U, R) = 0.
e R) = 0 proceeds in exactly the same way. Let a = (a ) be a 2-cocycle.
The proof that H 2 (U,
ijk
e j < k < l, we have
So for any four indices i <
(2)
ajkl − aikl + aijl − aijk = 0.
Define for i < j
bij =
X
θl alij
l
with the same conventions as above - for example, alij := −ailj if i < l < j. Then
X
(3)
(db)ijk =
θl (aljk − alik + alij )
l
and one checks again that the cocycle condition (2) implies that this term in parentheses at the
right of (3) is always equal to aijk . For example, if i < l < j < k, then
aljk − alik + alij = aljk + ailk − ailj = aijk
where the first equality holds by our convention and the last by the cocycle condition. This shows
that db = a, whence [a] = 0 in H 2 (U, R).
e
6
SECTION 7: PRESHEAVES AND ČECH COHOMOLOGY
A variant of Čech cohomology. As before, let X be a topological space and let A be a
presheaf on X. For an open cover U = {Ui }i∈I we have defined a cochain complex C ∗ (U, A) using
a linear order on the index set I: indeed, C n (U, A) is the product of all A(Ui0 ...in ) where Ui0 ...in
is non-empty and i0 < . . . < in . One can of course define a similar complex without this ordering
condition,
Y
e ∗ (U, A) =
C
A(Ui0 ...in )
(i0 ,...,in )
where the product ranges over all (n + 1)-tuples i0 , . . . , in for which Ui0 ...in is non-empty. This is a
larger complex with much redundant information (the group A(Ui0 ...in ) now occurs (n + 1)! times),
hence less convenient for calculations, but it computes the same cohomology as we will now show.
e ∗ (U, A) → C ∗ (U, A) is
Proposition 13. For any presheaf A on X, the obvious restriction map C
a homotopy equivalence of cochain complexes.
Proof. It will be notationally convenient to work with the underlying chain complexes (as we will
do in the next section). Let Cn (U) be the free abelian group on the set of ordered sequences
en (U) be the free abelian group on all such sequences (still
i0 < . . . < in with Ui0 ...in 6= ∅, and let C
with Ui0 ...in 6= ∅, but no longer necessarily ordered). If f : Cn (U) → Cm (U) is any linear map with
the property that for any generator ξ = (i0 , . . . , in ), f (ξ) is a linear combination of subsequences of
ξ, then f induces a map
f ∗ : C m (U, A) → C n (U, A),
by composition with f P
and restriction in the presheaf A.
Explicitly, if f (ξ) = ξi then Uξ ⊆ Uξi for each i, and we can define
X
f ∗ (a)ξ =
(aξi |Uξ ) ∈ A(Uξ ).
e∗ (U) and C
e ∗ (U, A). Let us call maps f with this property proper. For
A similar remark applies to C
example, the boundary ∂ : Cn (U) → Cn−1 (U), defined on generators by
∂(i0 , . . . , in ) =
n
X
(−1)k (i0 , . . . , iˆk , . . . , in ),
k=0
e n−1 (U, A) →
induces the Čech coboundary d = ∂ ∗ : C n−1 (U, A) → C n (U, A); and similarly for C
e n (U, A).
C
e
e
Now let j : Cn (U ) → C(U)
be the obvious inclusion, and let r : C(U)
→ Cn (U) be the map
(−1)|σ| (iσ(0) , . . . , iσ(n) ), if σ(i0 ) < . . . < σ(in );
r(i0 , . . . , in ) =
0,
if the i0 , . . . in are not all distinct.
(Thus, σ is the permutation putting (i0 , . . . , in ) in the right order, and |σ| is its sign). Note that
rj = id, and that j is obviously a chain map. In fact, r is a chain map too, as we will leave for you
to verify. Moreover, r and j are obviously both proper. We will prove that jr is chain homotopic
to the identity through a homotopy by proper maps
en (U) → C
en+1 (U),
hn : C
n ≥ 0.
SECTION 7: PRESHEAVES AND ČECH COHOMOLOGY
7
e n+1 (U, A) → C
e n (U, A) will form a cochain homotopy between the
Then the induced maps h∗n : C
∗ ∗
∗
∗
identity and r j = (jr) , proving that r and j ∗ are mutually inverse up to homotopy:
j∗
e ∗ (U, A) o
C
r
/
C ∗ (U, A).
∗
We will define the hn by induction, with the homotopy property
en (U) → C
en (U)
(4)
∂hn + hn−1 ∂ = jr − id : C
e0 (U) → C
e0 (U)). The definition of the hn will make use of a concatenation
(and ∂h0 = jr − id : C
en (U) → C
en+1 (U) by
map: for a fixed “basepoint” b ∈ I, we can define a map b ∗ (−) : C
(b, i0 , . . . , in ), if Ubi0 ...in 6= ∅;
b ∗ (i0 , . . . , in ) =
0,
otherwise.
note that if ξ = (i0 , . . . , in ) is a sequence with Ubξ 6= ∅ then
(5)
b ∗ ∂ξ + ∂(b ∗ ξ) = ξ.
(So if we wouldn’t be bothered with empty intersections, b∗ would be a homotopy from the identity
map to the zero map.)
Now let us define the hn . Since jr = id in degree 0, we let h0 = 0. Suppose that h0 , . . . , hn−1
have been defined in a way satisfying the homotopy relation (4) up to that stage. Define hn on a
generator ξ = (i0 , . . . , in ) by
(6)
hn (ξ) = i0 ∗ (jr(ξ) − ξ − hn−1 (∂ξ)).
Notice that since j and r are chain maps, the homotopy relation for hn−1 and hn−2 gives ∂(jr(ξ) −
ξ − hn−1 (∂ξ)) = hn−2 (∂∂ξ) = 0. So the formula (5), which is applicable to the right hand side of
(6) since all maps involved are proper, yields
∂hn (ξ) = jr(ξ) − ξ − hn−1 (∂ξ)
which is the homotopy equation (4) for the next step in the induction. Since hn as defined is
obviously proper, this completes the proof.