Let be a 2-manifold (a surface). We will associate to a vector space V over Z2 and a non-degenerate
bilinear symmetric form , . Just to remind you: bilinear means av + bv , w = a v, w + b v , w and
likewise for the second entry; symmetric means v,
Math 132 Notes 4
I didnt take notes last time for those who werent there, it was mostly a question-and-answer section.
In particular, we discussed
why every space cant be a trivial or 0-dimensional manifold it can as a set, but its topology will be
Math 132 Notes 2
Today were talking about embeddings.
Denition 1. f : X Y smooth is an embedding if it is an injective immersion that is also proper (the
preimage of a compact set is still compact).
Question: Given k 0 whats the smallest N for which X k e
Math 132 Notes 6
We wanted to say that transversality was generic. We proved that it was stable last time, which is
something like the set of maps f : X Y transversal to Z Y being an open subset of Map(X, Y ) (suitably
topologized so that homotopies are p
Math 132 Notes 5
Last time, we looked at transversality. If f : X Y is a smooth map and Z a submanifold of Y , then
if f Z, then f 1 (Z) X is a submanifold. If Z is of codimension m in Y , then f 1 (Z) is of codimension
m in X. Recall the denition of tran
Math 132 Notes 3
Today, well prove Sards Theorem:
Theorem 1 (Sards Theorem). If f : M N is smooth and C M is its set of critical points, then f (C) has
Its helpful to know a few things about measure zero for this proof. For example:
We dene the Euler characteristic of a space X, (X), to be the self-intersection number I(, ),
where is the submanifold cfw_(x, x) of X X. This might seem a bit arbitrary, but as well see, its
quite important. In fact, you may have heard of another
Recall that we are studying the Z2 -vector space V = cfw_f : M 1 : M closed, compact, connected/cobordism.
We wrote cobordism classes as [f ], and had [f ] + [g] = [f g]. We established the Moving Lemma: if
Z is dieomorphic to S 1 and I2 (f, Z) =
Today, we are talking about surgery. This isnt medical surgery, though I did have a creepy dream once
where I did it on my own leg. Consider a 2-dimensional manifold with an embedded S 1 whose normal space
is trivial (that is, dieomorphic to S 1 R
We now have a mechanism to calculate (V, , ) for pretty much every surface. Lets do the Klein bottle.
Below, Ive shown the Klein bottle, with surgery performed at the cycle e1 .
Clearly, the resulting surface is S 2 . By the theorem we proved las
Math 132 Notes
Underlying idea: work locally, think globally. Consider vector spaces for comparison. Every (nitedimensional real) vector space is isomorphic to Rn once we pick a basis, so we do all calculations under a
basis. But we want to develop the th