Problems for Tuesday, October 13: January 2009, II-3
August 2008, I-1, I-2, II-1
And
these
leftover problems.
Problems for Thursday, October 15: August 2008, I-3, I-4, II-2 and II-3
Problems for Tuesday, October 20: August 2008, I-5, I-6, II-4 and II-5.
And
these
leftover problems (I added another hint).
Problems for Thursday, October 22: August 2008, II-6
January 2008, I-1, I-2, II-1
Problems for Tuesday, October 27: January 2008, I-3, I-4, II-2, II-3
Problems for Thursday, October 29: January 2008, I-5, I-6, II-4, II-5
Problems for Tuesday, November 3: January 2008, II-6
August 2007, I-1, I-2, II-1
Problems for Thursday, November 5: August 2007, I-3, I-4, II-2, II-4
Also: finish January 2008, II-6.
Problems for Tuesday, November 10: August 2007, I-5, I-6, finish
II-2(b), II-3, II-4
I regret that I do not see how to do August 2007, I-3, unless the given
semigroup S has the cancellation property. If it does, then, as in
class, for each
p,q in G we construct r in G with rq=p. In particular, we
have, for any x in G, an e_x in G with e_x x = x. For any
x,y, e_x xy = e_y xy = xy, so e_x=e_y is the identity of G. Call
it e. For each q in G, we have r in G with rq=e, so we have
inverses too.
I believe that the committee made a mistake in not including the
cancellation property in the hypothesis, and so will not ask you
to think about what happens without it.
Problems for Thursday, November 12: August 2007, II-5, II-6,
January 2007, I-1, II-1
Please note: On the morning of Tuesday, December 15 (during exam
week), Professor
Zinger has kindly agreed to go over the topology problems from 2006. If anyone
has a problem with this, please e-mail me before our next class on
Tuesday.
Problems for Tuesday, November 17:
Finish August 2007, II-5. The hypothesis that f preserves the vertical
fibration means that f(x,y) = (u(x),v(x,y)).
Also: January 2007, I-2, I-3, II-2, II-3
Problems for Thursday, November 19:
January 2007, I-4, I-5, II-4, II-5
