110.420 TOPOLOGY OF SURFACES March 1996

Mid-Term Examination

THIS IS A TAKE-HOME EXAMINATION.

It is due back on or before WEDNESDAY, APRIL 3.

Allow ONE block of time, two hours suggested maximum.

This is an OPEN BOOK examination: you may consult the course textbook as often as you wish. You are not allowed to consult anybody else or any other book.

You may use any method covered in the course so far.

40 points total, allocated as indicated.

Partial credit may be available, but only if you show your working.

Use only the blue books provided.

Begin each of the four questions on a fresh page and number it clearly in the margin.

1. [5 points] Show that the cartesian product of two regular spaces is regular.

2. [10 points] Given a surface M with boundary ∂M, its double W is obtained from the disjoint union of two copies of M by gluing together the two copies of ∂M (precisely: identify the two copies of each point of ∂M). What is W if M is:

(a) an orientable surface of genus g with k open disks removed?

(b) a non-orientable surface of genus g with k open disks removed?

3. [5 points] Let X and Y be metric spaces with metrics d_{X} and d_{Y}. Define two metrics on the cartesian product X × Y by

d((x_{1}, y_{1}), (x_{2}, y_{2})) = max(d_{X}(x_{1}, x_{2}), d_{Y} (y_{1}, y_{2}))

and

d′((x_{1}, y_{1}), (x_{2}, y_{2})) = d_{X}(x_{1}, x_{2}) + d_{Y} (y_{1}, y_{2})

Show that they are equivalent (define the same topology on X × Y). (In fact, they both define the usual product topology.)

4. [20 points] A surface S (without boundary) is obtained from an octagon (its planar diagram) by gluing pairs of edges together according to a given scheme (as in Ex. 4.17). In each of the five cases below, classify the surface S if the gluing scheme is:

(a) abca^{−1}db^{−1}c^{−1}d^{−1};

(b) abca^{−1}db^{−1}d^{−1}c;

(c) abca^{−1}db^{−1}c^{−1}d;

(d) abcc^{−1}b^{−1}dd^{−1}a^{−1};

(e) abca^{−1}db^{−1}d^{−1}c^{−1}.

THIS QUESTION SHEET MAY BE RETAINED JMB