__Homework for Geometry and Topology 576a__

Some solutions: 1, 2, 3, 4, 5, 6, 7, 8, 9/10, 11/12.

The first few
assignments are a selection of problems to make sure that your basic point-set
topology is snappy. Then we move to more geometric material – the
classification of surfaces, and then homotopy type,
covering spaces and the fundamental group. We hope to get to a rudimentary
introduction to homology. Keep
your eye on this page, as dates will change and some more problems will be
added.

** ASSIGNMENT #1**: (due date: Friday, Sept 16)

(topological space,
subspaces, closures)

§13: 5,7,8

§16: 1,2,10

§17: 2,3,5,6,9,11,12,13,19

** ASSIGNMENT #2**: (due date: Friday, Sept 23)

(continuous, product,
metric)

§18: 2,13

§19: 1,2,3,7

§20: 2,4,8

§21: 2

** ASSIGNMENT #3: **(due date: Friday, Sept 30)

(connectedness, components)

§23: #5, #7, #11.

§24: #11, and also: Show that R^1 is not homeomorphic
to R^n for n>1,

§25: #4.

§26: #5.

** ASSIGNMENT #4:** (due
date: Friday, Oct 7)

(quotient spaces, compactness, separation axioms)

§22 : #2,#3.

§27: 6.

§28: #3abc.

§29: #8

§31: #5,
and: Show that every locally compact Hausdorff space
is regular.

** ASSIGNMENT #5:** (due
date: Friday, Oct 14)

(cell complexes)

1) Describe
cell structures for the closed orientable connected surfaces of genus 1 and
genus 2,

Draw pictures of the one-skeleton inside the
surface.

Relate
these to various ways of obtaining these surfaces by gluing together pairs of
sides of polygons.

2) Prove that a cell-complex is hausdorff.

3) Prove that a finite cell complex (i.e. finitely
many cells) is compact.

4) Explain
how to obtain any closed orientable surface by gluing together opposite sides
of polygons.

What
happens when you glue opposite sides of a hexagon?

Can you do
this in the non-orientable case too?

5) Consider
all the ways of identifying all sides of a triangle.

Determine which of these ways yield
homeomorphic spaces.

6) Let S be
a genus two surface embedded in the usual way in
3-dimensional space.

Let H be
the solid “handlebody” bounded by S. (H looks like a
solid pretzel with two holes.)

a) Give a cell structure for H.

b) Describe a simplicial complex
homeomorphic to H.

7) Describe
a simplicial complex homeomorphic to the 3-sphere.

8) Let B
and C be bouquets of two and three circles respectively. (So they are graphs
with one vertex and 2 and 3 edges respectively). Describe a cell structure for BxC.

9) Classify
the following connected surfaces according to genus, orientability,
and number of bounding circles

Note that
the genus of a surface with boundary, is defined to be
the genus of the corresponding closed surface obtained by attaching a disk
along each boundary circle.

http://www.math.mcgill.ca/wise/courses/576a/576a05HW_files/576aSurfaces.jpg

** ASSIGNMENT #6: **(due date: Friday, Oct 28)

(homotopy, fundamental group, begin
covering spaces)

§51: 3

§52: 1,4,5

§53: 4,6

§54: 5,7

** ASSIGNMENT #7: **(due date: Friday, Nov 4)

(Covering spaces, automorphisms)

1) Find a covering
space of a bouquet of 2 circles, whose automorphism group
is isomorphic to the alternating group of degree 4 (it has 12 elements)

2) Find two regular
covers of a bouquet of 2 circles which are different

but whose automorphism
groups are Z_6.

Find an infinite nonregular cover whose automorphism
group is Z_6.

3)
Find an infinite degree connected covering space of a bouquet of circles, with a
nontrivial

finite automorphism
group.

4)
Show that every finite group is the automorphism group
of some infinite degree covering

space of a bouquet of circles.

** ASSIGNMENT #8: **(due date: Friday, Nov 11)

(retractions, homotopy type)

§55: 1,2,3

§58: 1,2,5

(1) Prove that there is no retraction map from the
moebius strip to its boundary.

(2) Let S denote a genus 2 surface with two
points removed.

Use a sequence of pictures to describe a
deformation retraction from S to a graph.

(Start by drawing a picture of the graph in S.

(3) Let Z denote the z-axis in R^3, and let C
denote the unit circle in the x-y-plane.

Let M = R^3-C-Z. Describe a deformation
retraction from M to a torus.

** ASSIGNMENT #9: **(due date: Friday, Nov 18)

(UNIVERSAL
COVERS: )

(0) §79: 2,4. §81: 1,4.

(1)
Let X denote the standard 2-complex
of < b, c | b^2, c^3 >. Sketch a picture of the universal cover

of X.

(2)
Sketch a picture of the universal cover of X
where X is:

(a)
the union of a circle S with the unit disk D, where the point (1,
0) in S identified with the point (0, 0) in D.

(b)
the 1-skeleton of a tetrahedron.

(c)
the 1-skeleton of a cube.

(d)
the union of two tori identified at a single point.

(e)
the union of a torus and a circle identified at a single
point.

(3) Let X
denote the standard 2-complex of <a,b,c,d
| aba^-1b^-1cdc^-1d^-1> so X is a genus 2 surface.

Draw a
tiling of the plane by (combinatorial) pentagons with four around each vertex.

(a few layers suffice to get the point across.)

Doing this
in a sensible way, what happens to the sizes of the polygons in the outer
layers?

Now begin
to draw a tiling of the (hyperbolic) plane by octagons, eight around a vertex.

Label the
edges to indicate how this is the universal cover of the surface X above.

Use your
intuition in the (5,4) case to help imagine the (8,8)
case.

A)
Some problems about surfaces

1)
Let T be a torus with the usual cell structure obtained
by identifying opposite sides of a

square. Orient and label the vertical
and horizontal 1-cells of T by
v and h. Note that

any covering space of T
has an induced cell-structure. Draw the 1-skeleton of each
degree-3

connected cover of T (up to isomorphism) and label and orient its edges
to indicate the

covering map b: T → T.

2)
Consider an orientable genus 5 surface, with four boundary circles. Which other
surfaces

does it cover?

3)
Can the fundamental group of a genus 2 surface embed in the fundamental group of
a genus 1

surface?

** ASSIGNMENT #10: **(due date: Friday, Nov 25)

(Seifert
van Kampen

(0)
Find a basis for the fundamental group of the 1-skeleton
of a 3-dimensional cube.

(1)
Let M3 = M1#M2 denote the
connected sum of two connected 3-manifolds M1, M2.

Prove
that _{}pi_1(M_3) = _{}pi_1(M_1) * pi_1(M_2)

(2)
Let L1, . . . ,Lk denote disjoint lines in R3. Let L be the union of the Li. Let M = R3 − L.

What
is _{}pi_1M?

(3)
Let T1 and T2 be tori, and let C1 and C2 be simple closed curves in T1 and
T2 such that Ti−Ci

is connected. Consider the quotient space X = (T1 U T2)/(C1 = C2) obtained by identifying T1

and T2 along these circles. Find a presentation for pi_1X.

(4)
Draw a picture of an embedding of a closed genus 2-surface S in
R^3 such that the set of points

in S with maximal 3-rd coordinate is homeomorphic to a bouquet B of 2-circles.

Let
C be the cone on B, and let D
= S U_B C
be obtained by identifying C
and S along
B.

Compute
a presentation for _{}pi_1D.

(5) Show that the
figure 8 knot complement is not homeomorphic to either the trefoil or the unknot.

(Hint, consider
finite quotients of their fundamental groups using presentations)

** ASSIGNMENT #11: **(due date: Friday, Dec 2)

(Simplicial Homology computations.)

(1) Compute the homology groups of a
3-simplex.

(2) Compute the homology groups of the
2-torus.

(3) Compute the homology groups of the moebius strip and show that they are isomorphic to the
homology groups of the circle.

(4) Compute the homology groups of the klein bottle with Z coefficients
and with Z_2 coefficients.

(5) Let A and B be connected simplicial
complexes. Describe the homology groups of the wedge AvB
of A and B along a single point in terms of the homology of A and B.

** ASSIGNMENT #13: **(due date: Friday, Dec 9)

(Simplicial Homology computations.)

(1) Compute the homology groups of the
standard 2-complex of <a,b,c,d | abba, badcab, aca^-1c^-1 >

(2) Compute the homology groups of the
3-torus.

(3) Compute the homology groups of the
standard 2-complex of <a|a^6> with Z coefficients and with Z_p coefficients (for each p).

(4) C_1, C_2, p_1, p_2 denote two
circles and two points in R^3 (all of which are disjoint from each other).

Let X equal
R^3-C_1, C_2, p_1, p_2. Compute the homology groups of X.

(5) Draw pictures of the orientable pseudomanifolds representing generators of each of the

homology
groups computed in problem (4)