Introduction (Lecture 1)
February 3, 2009
One of the basic problems of manifold topology is to give a classication for manifolds (of some xed
dimension n) up to dieomorphism. In the best of all possible worlds, a solution to this problem would
provide the
Dieomorphisms and PL Homeomorphisms (Lecture 6)
February 16, 2009
Let M be a smooth manifold. In the previous lectures, we showed that M admits a Whitehead compatible triangulation, so that we can regard M as having an underlying piecewise linear manifold
Triangulation in Families (Lecture 7)
February 17, 2009
In the last lecture, we introduced the diagram of simplicial sets
ManmL ManmD Manm .
P
P
sm
Our goal in this lecture is to prove that the map is a trivial Kan bration. In other words, we wish to
show
Uniqueness of Triangulations (Lecture 5)
February 13, 2009
Our goal in this lecture is to prove the following result:
Theorem 1. Let M be a smooth manifold, and suppose we are given a pair of PD homeomorphisms f : K
M and g : L M . Then there exist PD ho
Existence of Triangulations (Lecture 4)
February 10, 2009
In the last lecture, we proved that if M is a smooth manifold, K a polyhedron, and f : K M a piecewise
dierentiable homeomorphism (required to be an immersion on each simplex), then K is a piecewis
Piecewise Linear Topology (Lecture 2)
February 8, 2009
Our main goal for the rst half of this course is to discuss the relationship between smooth manifolds
and piecewise linear manifolds. In this lecture, we will set the stage by introducing the essentia
Whitehead Triangulations (Lecture 3)
February 13, 2009
In the last lecture, we cited the theorem of Cannon-Edwards which shows that the 5-sphere S 5 admits
bad triangulations: that is, S 5 can be realized as the underlying topological space of polyhedra w
PL vs. Smooth Fiber Bundles (Lecture 8)
March 16, 2009
Our goal in this lecture is to begin to prove the following result:
Theorem 1. Suppose given a commutative diagram
K
f
/M
p
q
/N
L
where K and L are polyhedra, M and N are smooth manifolds, and the ho
Some Engulng (Lecture 9)
February 22, 2009
Our goal in this lecture is to complete the proof that every Whitehead triangulation of a smooth ber
bundle yields a PL ber bundle. Recall that we had reduced ourself to the case where the smooth ber
bundle in qu
The Kister-Mazur Theorem (Lecture 14)
March 6, 2009
Our rst goal in this lecture is to nish o the proof of the Kister-Mazur theorem, which guarantees that
the theory of microbundles is equivalent to the theory of Rn -bundles. We will work in the PL settin
Smoothings and Microbundles (Lecture 15)
March 11, 2009
We now return to the problem of smoothing piecewise linear manifolds. Recall the diagram
ManmL ManmD Manm
P
P
sm
of Lecture 6. We have shown that is a trivial Kan bration, so that we can also regard
Embeddings vs. Homeomorphisms (Lecture 13)
March 3, 2009
Our goal in this lecture is to carry out the main step in the proof of the Kister-Mazur theorem describing
the relationship between microbundles and Rn -bundles. Namely, we will prove the following:
Classifying Spaces for Microbundles (Lecture 12)
March 1, 2009
In this lecture, we will discuss construct a classifying space for microbundles of rank n. For simplicity,
we will restrict our attention to piecewise linear microbundles (since this will be t
Smoothing PL Fiber Bundles (Lecture 10)
February 25, 2009
Recall our assertion:
Theorem 1. Suppose given a commutative diagram
K
f
/M
p
q
/N
L
where K and L are polyhedra, M and N are smooth manifolds, and the horizontal maps are PD homeomorphisms. Assume
Microbundles (Lecture 11)
February 27, 2009
In this lecture, we will continue our study of microbundles. Recall that a microbundle over X is a map
p : E X equipped with a section s : X E . We will sometimes abuse terminology and simply refer to
p : E X or