Lecture 15: Duality
We ended the last lecture by introducing one of the main characters in the remainder of the
course, a topological quantum eld theory (TQFT). At this point we should, of course, elaborate
on the denition and give examples, background, m
Lecture 10: Thom spectra and X-bordism
We begin with the denition of a spectrum and its antecedents: prespectra and -prespectra.
Spectra are the basic objects of stable homotopy theory. We construct a prespectrumthen a
spectrumfor each unstable or stable
Lecture 9: Tangential structures
We begin with some examples of tangential structures on a smooth manifold. In fact, despite
the namewhich is appropriate to our application to bordismthese are structures on arbitrary
real vector bundles over topological s
Lecture 11: Hirzebruchs signature theorem
In this lecture we dene the signature of a closed oriented n-manifold for n divisible by four. It
is a bordism invariant Sign : SO Z. (Recall that we dened a Z/2Z-valued bordism invariant
n
of non-oriented manifol
Lecture 12: More on the signature theorem
Here we sketch the proof of Theorem 11.46. In the last lecture we indicated most of the techniques
involved by proving the theorem for 4-manifolds. There are two additional inputs necessary for the
general case. F
Lecture 14: Bordism categories
The denition
Fix a nonnegative1 integer n. Recall the basic Denition 1.19 of a bordism X : Y0 Y1 whose
domain and codomain are closed (n 1)-manifolds. A bordism is a quartet (X, p, 0 , 1 ) in which
X is a compact manifold wi
Lecture 13: Categories
We begin again. In Lecture 1 we used bordism to dene an equivalence relation on closed
manifolds of a xed dimension n. The set of equivalence classes has an abelian group structure
dened by disjoint union of manifolds. Now we extrac
Lecture 8: More characteristic classes and the Thom isomorphism
We begin this lecture by carrying out a few of the exercises in Lecture 1. We take advantage of
the fact that the Chern classes are stable characteristic classes, which you proved in Exercise
Lecture 7: Characteristic classes
In this lecture we describe some basic techniques in the theory of characteristic classes, mostly
focusing on Chern classes of complex vector bundles. There is lots more to say than we can do
in a single lecture. Much of
Lecture 2: Orientations, framings, and the Pontrjagin-Thom construction
Thoms great contribution was to translate problems in geometric topologysuch as the computation (Theorem 1.37) of the unoriented bordism ringinto a problem of homotopy theory. The
cor
Lecture 1: Introduction to bordism
Overview
Bordism is a notion which can be traced back to Henri Poincar at the end of the 19th century, but
e
th century in the hands of Lev Pontrjagin and Ren Thom [T]. Poincar
it comes into its own mid-20
e
e
originally
Lecture 3: The Pontrjagin-Thom theorem
In this lecture we give a proof of Theorem 2.35. You can read an alternative exposition in [M3].
We begin by reviewing some denitions and theorems from dierential topology.
Neat submanifolds
Recall the local model (1
Lecture 4: Stabilization
There are many stabilization processes in topology, and often matters simplify in a stable limit.
As a rst example, consider the sequence of inclusions
(4.1)
S 0 S 1 S 2 S 3
where each sphere is included in the next as the equato
Lecture 6: Classifying spaces
A vector bundle E M is a family of vector spaces parametrized by a smooth manifold M . We
ask: Is there a universal such family? In other words, is there a vector bundle E univ B such that
any vector bundle E M is obtained fr
Lecture 5: More on stabilization
In this lecture we continue the introductory discussion of stable topology. Recall that in Lecs
ture 1 we introduced the stable stem , the stable homotopy groups of the sphere. We show
s is a Z-graded commutative ring (Den