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.

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.

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 TiCi

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)