3. Wed, Jan. 22
Last time, we introduced a path-composition operation (concatenation). The path
rst travel along in double time, then travel along in double time.
Proposition 3.1. The above operation only depends on path-homotopy classes. That is, if
and
17. Mon, Feb. 24
Let F be a transitive right G-set. Then F H\G for some H G. We assume that B has
=
a universal cover q : X ! B. Recall that we showed in Theorem 11.4 that the group of deck
transformations of X is isomorphic to G.
Proposition 17.1. The ac
20. Mon, Mar. 3
Dashing through the snow, in a one-horse open sleigh. . .
21. Wed, Mar. 5
Exam Day!
Long time the manxome foe he sought
So rested he by the Tumtum tree,
And stood awhile in thought.
22. Fri, Mar. 7
Last time (week), we showed that if a spa
8. Mon, Feb. 3
Snow Day!
9. Wed, Feb. 5
Today, were going to calculate 1 (RP2 ), but rst I want to discuss a result about contractibility
of paths.
Proposition 9.1.
(1) Let 2 1 (X, x0 ). Then 'p cx0 if and only if : S 1 ! X extends
2 ! X.
to a map D
(2) L
14. Mon, Feb. 17
The interesting, new result here concerns the existence of lifts.
Proposition 14.1. (Lifting Criterion) Let p : E ! B be a covering and let f : Z ! B, with Z
very connected. Given points z0 2 Z and e0 2 E with f (z0 ) = p(e0 ), there is a
CLASS NOTES
MATH 651 (SPRING 2013)
BERTRAND GUILLOU
1. Wed, Jan. 15
Here are a list of main topics for this semester:
(1) homotopy, homotopy equivalence (Hatcher - Ch. 0, Ch. 1.1; Lee - Ch. 7)
(2) the fundamental group (topology
algebra) (Hatcher - Ch. 1.
29. Mon, Mar. 31
Last time we were discussing the torus, and we arrived at 1 (T 2 ) F (a, b)/haba
=
a proof that this is isomorphic to Z2 .
1 b 1 i.
Here is
Proposition 29.1. The natural map ' : F (a, b) ! Z2 dened by '(a) = (1, 0) and '(b) = (0, 1)
induc
23. Mon, Mar. 10
The free product has a universal property, which should remind you of the property of the
disjoint union of spaces X q Y . First, for any groups H and K, there are inclusion homomorphisms
H ! H K and K ! H K.
Proposition 23.1. Suppose tha
41. Mon, Apr. 28
We saw last time that homology interacts nicely with disjoint unions. We list here a few more
nice properties of homology, without proof.
Proposition 41.1.
(1) If X is a k-dimensional CW complex, then Hn (X) = 0 for all n > k.
(2) Let f :
35. Mon, Apr. 14
Ok, so we know that n (RP2 ) n (S 2 ). What are these groups? We will show later that
=
2 (S 2 ) Z. Just like for S 1 , a generator for this group is the identity map S 2 ! S 2 . But the
=
fascinating thing is that, in contrast to S 1 , t
32. Mon, Apr. 7
Last time, we introduced the genus g surfaces Mg , dened as the g-fold connected sum of copies
of T 2 . We found that
1 (Mg ) F (a1 , b1 , . . . , ag , bg )/[a1 , b1 ] . . . [ag , bg ].
=
Proposition 32.1. 1 (Mg )ab Z2g .
=
Proof. Let F =
26. Mon, Mar. 24
The next application is the computation of the fundamental group of any graph. We start
by specifying what we mean by a graph. Recall that S 0 R is usually dened to be the set
S 0 = cfw_ 1, 1. For the moment, we take it to mean instead S
38. Mon, Apr. 21
Example 38.1. Take X = T 2 . The standard cell structure we have used has a single 0, two 1-cells
a and b, and a single 2-cell e attached via aba 1 b 1 . Since there is a single 0-cell, this means that
automatically d1 = 0. To calculate d