Download Exercises in Algebraic Topology version of February

Exercises in Algebraic Topology
version of February 2, 2017
Exercise 1. Let (X, d), (Y, d) be metric spaces anf f : X −→ Y be a map.
Show that f is continuous (in the sense of metric spaces - δ − ε-definition)
if and only if for each open U ⊂ Y the set f −1 (U ) is open in X.
Exercise 2. Show that equivalent norms on a (real) vector space induce the
same topologies.
Exercise 3. Show that S1 is homeomorphic to
S11 := {(x0 , x1 ) ∈ R2 | |x0 | + |x1 | = 1}
and to
S1∞ := {(x0 , x1 ) ∈ R2 | max (|x0 |, |x1 |) = 1}.
Exercise 4. Show that S1 is homeomorphic to I/∂I, I = [0, 1], and that
RP n is homeomorphic to Sn / ± 1.
Definition 1. A continuous map f : X −
→ Y between topological spaces
is called an embedding, if f is a homeomorphism f : X −
→ f (X) onto its
image.
Exercise 5. Find an embedding of the Möbius strip M = I × I/ ∼ into R3 .
Recall that ∼ is the minimal equivalence relation such that (1, t) ∼ (0, 1 − t).
Exercise 6. Find an injective (continuous) map f : R −→ S1 × S1 which is
not an embedding. Hint: consider S1 ⊂ C and t 7→ eiαt for appropriate α.
Exercise 7. We consider X = R endowed with the following topologies:
τ1 := τstd , τ2 = τdisc , τ3 = τindisc = τtriv , τ4 = τcofinite , τ5 = τcocountable where
the latter two are defined as follows:
τcofinite = {U | X \ U finite }∪{∅}
τcocountable = {U | X \ U countable }∪{∅}.
Consider the sequences (xn )n∈N and (yn )n∈N defined through xn = n1 , yn =
(−1)n and determine whether they converge with respect to τi and if so, to
which limit(s), for i = 1, . . . , 5.
Exercise 8. Show that if f : X −→ Y is continuous, then it is sequentially
continuous. Does the converse hold? (hint: determine when a sequence
converges for the cocountable topology and compare to the discrete topology)
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EXERCISES IN ALGEBRAIC TOPOLOGY VERSION OF FEBRUARY 2, 2017
Exercise 9. Show that if a space X is second countable then it is first
countable. Show that if a space X is second countable then it contains a
dense subset which is countable. Show that if X is a metric space (hence in
particular first countable) which contains a dense countable subset A ⊂ X,
then it is second countable.
Exercise 10. Show that singletons (i.e. sets {p} with one element) are
closed in Hausdorff spaces. Also prove that X being Hausdorff is equivalent
to the diagonal being closed in the product X × X.
Exercise 11. Show that if A ⊂ X is closed and (xn )n∈N is a sequence in A
that converges to a limit x ∈ X, then x ∈ A.
Exercise 12. Show that if X is a locally compact space, then given a point
p ∈ X and an open U ⊂ X with p ∈ U , then one can find an open V ⊂ U
containing p such that the closure V̄ in X is compact and still V̄ ⊂ U .
Exercise 13. Show that a closed subset of a compact space is compact. Show
that a compact subset of a Hausdorff space is closed.
Hint: for the second part you could start like this. Let p be a given point
in the complement of the compact set. For every point q in the compact set
separate p from q by the Hausdorff property. Then use compactness.
Exercise 14. Let f : X −→ Y be a continuous map. Assume that X is
(quasi-)compact. Show that f (X) ⊂ Y is (quasi-)compact.
Exercise 15. Let f : X −→ Y be a continuous map from a quasi-compact
space X to a Hausdorff space Y . Show that f (X) ⊂ Y is closed.
Exercise 16. Let X be a locally compact (Hausdorff ) space and X∞ :=
`
X {∞} be its one-point (or Alexandrov) compactification. Recall that the
open sets were defined as the open sets in X or the subsets U ⊂ X∞ so that
X∞ \ U ⊂ X is compact. Show that this defines a topology which makes X∞
a compact space. Show that X ⊂ X∞ is dense if X is not itself compact.
What happens if X is compact?
Exercise 17. Let f : X −→ Y be a bijective continuous map. Assume that
X is compact and that Y is Hausdorff. Show that f is an homeomorphism.
Can you give an example of a quasi-compact space X, a Hausdorff space Y
and a continuous bijective map f : X −→ Y which is not a homeomorphism?
Exercise 18. Let f : X −→ Y be a continuous map between topological
spaces and suppose that X is connected. Show that f (X) ⊂ Y is connected.
Exercises in Algebraic Topology
version of February 2, 2017
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Exercise 19. Prove the intermediate value theorem from elementary analysis using the notion of connectedness: if f : [a, b] −→ R is a continuous
function, then f takes any value between f (a) and f (b). Without loss of
generality asume f (a) < f (b).
Exercise 20. A subset S ⊂ Rn is called star-shaped (sternförmig), if there
is p ∈ S such that for all q ∈ S and for all t ∈ [0, 1] we have tp+(1−t)q ∈ S.
Show that a star-shaped subset S ⊂ Rn is pathwise connected (and hence
connected).
Exercise 21. Let X be a topological space and let a, b ∈ X. We write a
for a path from a to b. Show that the subsets
X ⊃ Xp := {q ∈ X | ∃p
b
q}
are the equivalence classes for an equivalence relation ∼. How would you
define a ∼ b?
Exercise 22. Consider the so-called “nodal curve” C = {(x, y) ∈ R2 |
x2 + x3 = y 2 }. Draw a picture of C ⊂ R2 . Show that C is not locally
Euclidean. You can e.g. look at neighborhoods of (0, 0) ∈ C. How do
connectedness properties change if you pass to the complement of (0, 0)?
Exercise 23. Let N := (1, 0, . . . , 0), S := (1, 0, . . . , 0) ∈ Sn be “north and
south pole” on the unit sphere. Show that stereographic projection defined by
1
ϕN : Sn \ N −→ Rn , x = (x0 , . . . , xn ) 7→ 1−x
(x1 , . . . , xn ) defines a chart for
0
n
S and A = {ϕN , ϕS } defines an atlas where ϕS is defined similarly. Show
that this is a smooth (i.e. C ∞ ) atlas which defines the same differentiable
structure on Sn as the atlas defined in the lecture. Can you give an atlas on
the sphere with one chart only?
Exercise 24. Show that the dimension of a connected manifold X is constant. Thus, we may write dim X ∈ N.
Exercise 25. Show that metric spaces are normal, i.e. that Every two
disjoint closed subsets can be separated by open subsets.
Exercise 26. Write down a proof for the following two facts:
(1) The boundary ∂X of a manifold X with boundary is a closed subset
X.
(2) The boundary ∂X of an n-dimensional C k -manifold X with boundary is in a natural way an (n − 1)-dimensional closed C k -manifold.
Exercise 27. Show that being homotopic (also relative a subset A ⊂ X) is
an equivalence relation on continuous maps X −→ Y .
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EXERCISES IN ALGEBRAIC TOPOLOGY VERSION OF FEBRUARY 2, 2017
Exercise 28. Complete the proof of the theorem saying that the fundamental
group is a group.
Exercise 29. Show that for a path-connected space X and points x0 , x1 ∈ X
every path from x0 to x1 induces an isomorphism π1 (X, x0 ) −→ π1 (X, x1 ).
Exercise 30. Show that the projections to each factor induce an isomorphism π1 (X ×Y, (x0 , y0 )) −→ π1 (X, x0 )×π1 (Y, y0 ) for pointed spaces (X, x0 ),
(Y, y0 ).
Exercise 31. A space X is called contractible1, if it is homotopy equivalent
to a one point space. Show that every topological R-vector space (i.e., an
R-vector space with a topology such that addition and scalar multiplication
are continuous) is contractible. Conclude that Rn is contractible.
Exercise 32. Show that R and [0, 1) are homotopy equivalent.
Exercise 33. Let s ∈ S1 and denote by 1n×n the (n × n)-unit matrix. Show
that π1 (S1 , s) is isomorphic to π1 (GL2 (R), 12×2 ).
Hint: Use Gram-Schmid.
Exercise 34. Use the universal property of the free product G1 ∗G2 to verify
the universal property of the pushout G1 ∗G0 G2 .
S
Exercise 35. Let X = i∈I Ui be a cover and let γ : I −→ X be a path
in X. Show that there is N ∈ N such that for all n = 1, . . . , N there is
n
i = i(n) ∈ I such that γ([ n−1
N , N ]) ⊂ Ui(n) .
S
Exercise 36. Let X = i∈I Ui be a cover and let H : I × I −→ X be
continuous. Show that there is N ∈ N such that for all n, m = 1, . . . , N
n
m−1 m
there is i = i(n, m) ∈ I such that H([ n−1
N , N ] × [ N , N ]) ⊂ Ui(n,m) .
Exercise 37. Let H : I × I −→ X be a continuous map and define paths
F0 , F1 , G0 , G1 : I −→ X by
F0 (t) = H(0, t),
F1 (t) = H(1, t),
G0 (t) = H(t, 0),
G1 (t) = H(t, 1).
Show that G1 ∗ F0 '∂I F1 ∗ G0 .
Exercise 38. Let C× := C \{0} and let π : C× −→ C× be the function
π(z) = z n . Show that π is a covering space.
Exercise 39. Let Y be a Hausdorff space and let π : X −→ Y be discrete
(:⇔ π −1 (y) discrete for all y ∈ Y ), open, proper (:⇔ π −1 (K) compact for
1zusammenziehbar auf Deutsch.
Exercises in Algebraic Topology
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every compact set K ⊂ Y ), and unramified (:⇔ locally injective). Show that
π is a covering space if Y is connected.
Hint: you can for example proceed as follows.
(1) Show that X is Hausdorff.
(2) Use discreteness and properness of π to deduce that all fibers Fy :=
π −1 (y), y ∈ Y are finite. You probably need to take into account
Exercise 10.
(3) Show that for a given y ∈ Y and x ∈ Fy there are open neighborhoods
Ux ⊂ X of x in X such that the {Ux | x ∈ Fy } are pairwise disjoint.
(4) Show that for each y ∈ Y there is an open neighborhood U ⊂ Y of y
∼ U × Fy .
such that π −1 (U ) =
(5) Show that π is a covering space.
Exercise 40. Let Y be a path connected space and let f : (X, x0 ) −→ (Y, y0 ),
f 0 : (X 0 , x00 ) −→ (Y, y0 ) be morphisms of pointed spaces such that f and f 0
are universal covers. Show that there is a unique isomorphism g : (X, x0 ) −→
(X 0 , x00 ) such that f = f 0 ◦ g.
Exercise 41. Complete the proof of the existence of a universal covering in
the lecture of January 9. Let Y be path connected, locally path connected,
and semi-locally simply connected. Denote W Y the set of all paths in Y
starting at y0 ∈ Y and let X := W Y / ∼ where we identify paths with the
same endpoint if they are homotopic relative ∂I. Show:
(1) For an open set U ⊂ Y and γ ∈ W Y consider
hγ, U i := {[u ∗ γ] | u : I −→ U, u(0) = γ(1)}.
Show that the sets of the form hγ, U i form a basis of a unique topology
on X. You may show that hγ, U1 ∩ U2 i ⊂ hγ, U1 i ∩ hγ, U2 i.
(2) Show that the canonical projection f : X −→ Y is continuous with
respect to the topology just defined.
(3) Show that f : X −→ Y is open.
(4) Show that f : X −→ Y is a covering space with typical fiber π1 (Y, y0 )
(endowed with the discrete topology) and that pathwise connected and
simply connected open sets U ⊂ Y are admissible. Why does every
point y ∈ Y have a pathwise connected and simply connected open
neighborhood?
Exercise 42. Let f : X −→ Y be a covering space, let y ∈ Y be a point, let
F := f −1 (y), and let G := π1 (Y, y) be the fundamental group with basepoint
y. For g = [γ] ∈ G and x ∈ F we define g.x := γ
e(1) where γ
e : I −→ X
is the unique lift of γ starting at x. Show that this defines an action of G
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EXERCISES IN ALGEBRAIC TOPOLOGY VERSION OF FEBRUARY 2, 2017
on F , which is transitive if and only if X is path connected. We call it the
monodromy action.
Definition 2. Let f : X −
→ Y be a covering space with typical fiber F . The
cardinality of F is called the number of sheets or the degree of the covering.
∼
=
In a trivialization ϕ : f −1 (U ) −−−→ U × F for an admissible U ⊂ Y the sets
ϕ−1 (U × {f }) for f ∈ F are called sheets. A covering is said to be n-sheeted
or n : 1 if F has cardinality n ∈ N.
Exercise 43. Let f : X −→ Y be a covering space with x0 ∈ X, y0 =
f (x0 ) ∈ Y . Assume that Y is path connected, locally path connected, and
semi-locally simply connected and that X is path connected. Show that the
degree of the cover is equal to the index of f∗ π1 (X, x0 ) in π1 (Y, y0 ). Can you
show the same result without the hypotheses on Y ?
Exercise 44. Let f : X −→ Y be a 2-sheeted covering. Show that f is
normal.
Exercise 45. Let f : C −→ C be a polynomial function. Show that there
is a finite set ∆ ⊂ C such that for X := C \ f −1 (∆) and Y := C \ ∆ the
restriction f : X −→ Y is a covering space whose degree equals the degree of
the polynomial f .
Hint: Use exercise 39 to show that f : X −→ S is a covering space. Properness can be obtained by extending f to a map CP 1 −→ CP 1 (which will not
be a covering space) and use that continuous maps between compact spaces
are proper (why?). You may use the well-known result from complex analysis
saying that holomorphic maps C −→ C are open.
Exercise 46. Calculate a set ∆ ⊂ C as in Exercise 45 for the following
polynomials. Which defines a normal cover? Calculate the group of deck
transformations.
(1) f1 = z 3 − 5z 2 + 8z − 4,
(2) f2 = z 4 − 4z 2 + 7.
Exercise 47. Consider the polynomial f = 3z 2 − 2z 3 and the associated
covering f : X −→ Y as in Exercise 45. Denote by C(x) the field of rational
functions in one indeterminate. Show that the field extension C(x) −→ C(z)
defined through x 7→ f (z) is a degree 3 extension which is not Galois. Compute its Galois closure. What could be the “Galois closure” of f : X −→ Y ?
Exercise 48. Let f : X −→ Y be a normal cover and let γ : I −→ Y be
a closed loop at y0 ∈ Y . Denote for x ∈ F = f −1 (y0 ) the unique lift of γ
along f by γ
ex . Show that γ
ex is a closed loop for one x ∈ F if and only if γ
ex
Exercises in Algebraic Topology
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7
is a closed loop for every x ∈ F . Can you give an example of a non-normal
cover where this statement is false?
Exercise 49. Let f : (X, x0 ) −→ (Y, y0 ) be a normal cover, let X be path
connected, and let G := π1 (Y, y0 ). Show that the action of G on X through
deck transformations can be described as follows: given g = [γ] ∈ G and
x ∈ X, choose a path w : x0
x in X and let δ := w
e∗γ
e where γ
e is the
unique lift of γ starting at x0 and w
e is the unique lift of f ◦ w starting at
γ
e(1). Then g.x = δ(1).
Exercise 50. To practice some diagram chase, prove the 5-lemma: given
a diagram of R-modules (R a commutative ring with 1) with exact rows as
follows:
M1
ϕ1
N1
/ M2
/ M3
ϕ2
ϕ3
/ N2
/ M4
ϕ4
/ N3
/ N4
/ M5
ϕ5
/ N5
Show that
(1) if ϕ2 , ϕ4 are injective, and ϕ1 is surjective, then ϕ3 is injective.
(2) if ϕ2 , ϕ4 are surjective, and ϕ5 is injective, then ϕ3 is surjective.
(3) if ϕ1 , ϕ2 , ϕ4 , ϕ5 are isomorphisms, then so is ϕ3 .
Exercise 51. To practice some more diagram chase, prove the snake lemma:
given a diagram of R-modules (R a commutative ring with 1) with exact rows
as follows:
0
/ M3
/ M2
M1
ϕ1
/ N1
ϕ2
/ N2
/ 0
ϕ3
/ N3
Construct an exact sequence
ker ϕ1 −→ ker ϕ2 −→ ker ϕ3 −→ coker ϕ1 −→ coker ϕ2 −→ coker ϕ3 .
Exercise 52. Show that there is an exact sequence of reduced homology
groups
e p (A) −→ H
e p (X) −→ Hp (X, A) −→ H
e p−1 (A) −→ . . .
. . . −→ H
Exercise 53. Let
η
ε
0 −→ A0 −−→ A −−→ A00 −→ 0
be an exact sequence of abelian groups. Show that the following are equivalent:
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EXERCISES IN ALGEBRAIC TOPOLOGY VERSION OF FEBRUARY 2, 2017
(1) The sequence splits at ε, i.e., there is s : A00 −→ A such that s ◦ ε =
idA00 . The map s is called a section for ε.
(2) The sequence splits at η, i.e., there is s : A −→ A0 such that η ◦ s =
idA00 .
(3) A ∼
= A0 ⊕ A00 in such a way that η is a0 7→ (a0 , 0) and ε is (a0 , a00 ) 7→
a00 .
Exercise 54. Let
η
ε
0 −→ G0 −−→ G −−→ G00 −→ 0
be an exact sequence of groups. Show that the following are equivalent:
(1) The sequence splits at ε.
(2) G ∼
= G0 o G00 in such a way that η is g 0 7→ (g 0 , 0) and ε is (g 0 , g 00 ) 7→
g 00 .
Further show, that the following are equivalent:
(1) The sequence splits at η.
(2) G ∼
= G0 × G00 in such a way that η is g 0 7→ (g 0 , 0) and ε is (g 0 , g 00 ) 7→
g 00 .
Hint: Show that if the sequence splits at η via s : G00 −→ G, then for every
h ∈ G00 and g ∈ G0 there is a unique g̃ ∈ G0 such that η(g̃) = s(h)η(g)s(h)−1 .
Deduce that ϕh : G0 −→ G0 , g 7→ g̃ is an automorphism of G0 .
Definition 3. Let X be a topological space. A subset A ⊂ X is called a
retract (of X) if there is r : X −
→ A such that r|A = idA .
Exercise 55. Show that if A ⊂ X is a retract, then we have Hp (A) ⊂ Hp (X)
for all p. Deduce from a comparison of singular homology groups that Sn−1
is not a retract of Dn for n ≥ 1.
We use this to deduce Brouwer’s Fixed point theorem as follows: TBD
Invariance of the domain. TBD
Exercise 56. Give a CW-decomposition of Sn with two k-cells in every
dimension 0 ≤ k ≤ n. Use it to calculate singular homology for n = 1, 2, 3.
Exercise 57. Show that f : D2n −→ CP n , z = (z0 , . . . , zn−1 ) 7→ [z0 : . . . :
zn−1 : 1 − kzk] induces a homeomorphism D2n ∪f CP n−1 −→ CP n and that
consequently CP n has a CW-decomposition with one k-cell in every even
dimension and no odd dimensional k-cells.
Exercise 58. Calculate the degree of all possible continuous maps f : S0 −→
S0 .
Exercises in Algebraic Topology
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9
Exercise 59. Give a CW-decomposition of the 2-torus T = S1 × S1 and use
it to calculate its homology.
Exercise 60. Give a CW-decomposition of your favorite platonic solid and
count E − K + F where E stands for 0-cells, K for 1-cells, and F for 2-cells.
Exercise 61. Calculate TorZ1 (Z/m, Z/n) for m, n ∈ Z.
Exercise 62. Calculate Hp (M, Z/2Z) for all p where M is the Möbius strip.
What can you say about Hp (M, Z/3Z)?
Exercise 63. Show that TorR
→ K −→ F −→
1 (M, N ) is well defined: if 0 −
0
0
M −→ 0 and 0 −→ K −→ F −→ M −→ 0 are exact, show that there is an
isomorphism ker(K ⊗ N −→ F ⊗ N ) −→ ker(K 0 ⊗ N −→ F 0 ⊗ N ).
Hint: Show first that for all exact sequences 0 −→ K −→ F1 −→ F0 −→ 0
with free R-modules F0 , F1 we have ker(K ⊗ N −→ F1 ⊗ N ) = 0. Then
construct a morphism F 0 −→ F such that K 0 is mapped to K. Reduce to the
case where F 0 −→ F is surjective and proceed by diagram chase.
Exercise 64. Let C• be a chain complex such that all Hp (C• ) are R-modules
of finite rank for a ring R (commutative with 1) and suppose that only finitely
many of them are non-zero. We define the Euler characteristik to be
X
e(C• ) :=
(−1)i rkHi (C• ).
i∈Z
Show that if all Cp have finite rank and if only finitely many of them are
P
non-zero, then e(C• ) = ẽ(C• ) where ẽ(C• ) := i∈Z (−1)i rkCi .
Hint: Show that if 0 −→ A• −→ C• −→ B• −→ 0 is a short exact sequence of
chain complexes with the above finiteness conditions, then ẽ(C• ) = ẽ(A• ) +
ẽ(B• ). Show next that ẽ(C•+k ) = (−1)k ẽ(C• ).
Exercise 65. Let M, N be R-modules and recall the definition of Ext(N, M ):
this was the set of all equivalence classes of extensions of N by M where
two extensions 0 −→ M −→ E −→ N −→ 0, 0 −→ M −→ E 0 −→ N −→ 0 were
called equivalent if there is a morphism between them
(0.1)
0
0
/ M
/ M
/ E
/ N
/ 0
/ N
/ 0
φ
/ E0
such that both squares commute. Show that φ necessarily is an isomorphism
in this case. Show that an extension is split if and only if it is equivalent to
the trivial extension 0 −→ M −→ M ⊕ N −→ N −→ 0. Given two extensions
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EXERCISES IN ALGEBRAIC TOPOLOGY VERSION OF FEBRUARY 2, 2017
f
n
f0
n0
ξ := [0 −→ M −−→ E −−→ N −→ 0], ξ 0 := [0 −→ M −−→ E 0 −−→ N −→ 0],
we define their sum ξ + ξ 0 to be the extension
0 −→ M −→ F −→ N −→ 0,
where F = {(e, e0 ) ∈ E ⊕ E 0 | n(e) = n0 (e0 )}/{(f (m), 0) − (0, f 0 (m)) | m ∈
M } and M −→ F and F −→ N are the canonical maps (describe them!).
Show that this defines the structure of an abelian group on Ext(N, M ).