Algebraic Surgery (Lecture 11)
February 17, 2011
Before getting to the main topic of this lecture, let us pick up a few loose ends from the previous lecture.
Recall that for every A -ring R with involution, we dened stable -categories LModperf and LModfp
L-theory Spaces (Lecture 6)
February 3, 2011
Let C be a stable -category equipped with a nondegenerate quadratic functor Q : Cop Sp, which
we regard as xed throughout this lecture. We let B : Cop Cop Sp denote the polarization of Q, and
D : Cop C the corr
Simplicial Spaces (Lecture 7)
February 6, 2011
Let C be a stable -category equipped with a nondegenerate quadratic functor Q : Cop Sp. In the last
lecture, we dened an L-theory space L(C, Q), whose path components comprise the abelian group L0 (C, Q)
of L
L Groups (Lecture 5)
February 2, 2011
Let C be a stable -category equipped with a quadratic functor Q : Cop Sp. The polarization B of Q
is a symmetric bilinear functor on C. We will say that Q is nondegenerate if B is nondegenerate: that is, if
there is a
Quadratic Functors (Lecture 4)
February 15, 2011
In this lecture, we will introduce the notion of a quadratic functor Q on a stable -category C, and dene
the L-group L0 (C, Q). We begin with a short review of the classical theory of quadratic forms.
Denit
Categorical Background (Lecture 2)
February 2, 2011
In the last lecture, we stated the main theorem of simply-connected surgery (at least for manifolds of
dimension 4m), which highlights the importance of the signature X as an invariant of an (oriented) P
Stable -Categories (Lecture 3)
February 2, 2011
In the last lecture, we introduced the denition of an -category as a generalization of the usual notion of
category. This denition is one way of formalizing the notion of a higher category in which all k -mo
Localization (Lecture 8)
February 10, 2011
Let C be a stable -category equipped with a nondegenerate quadratic functor Q : Cop Sp. In the last
lecture, we asserted without proof that the simplicial space Poinc(C, Q) satises the Kan condition. Our
goal in
Proof of the Kan Property (Lecture 9)
February 11, 2011
Let C be a stable -category equipped with a nondegenerate quadratic functor Q : Cop Sp. Let B be
the polarization of Q and D the associated duality functor. Our goal in this lecture is to prove the f
Odd L-Theory of the Integers (Lecture 15)
February 28, 2011
Our goal now is to compute the quadratic L-groups Lq (Z). We have seen that the answers depend only
n
on the congruence class of n modulo 4. We therefore have four calculations to perform. We wil
The Even L-groups of Z (Lecture 16)
March 2, 2011
In this lecture we will compute the quadratic L-groups of Z in even dimensions. We begin by considering
Lq 2 (Z). Consider the pair (LModfp , 2 Qq ). Note that 2 Qq (Z)
2 (2 Z)h2 can be identied with
Z
the
L-Groups of Fields (Lecture 13)
February 23, 2011
Our goal in this section is to carry out some calculuations of L-groups in simple cases. We begin with
the following observation:
Proposition 1. Let R be an associative ring with involution. Then the L-gro
Surgery Below the Middle Dimension (Lecture 12)
February 17, 2011
In the previous lecture, we discussed the process of surgery. If C is a stable -category equipped with a
nondegenerate functor Q and we are given a quadratic object (X, q ), a map : X X , a
L-Theory of Rings and Ring Spectra (Lecture 10)
February 14, 2011
Let R be an associative ring. Recall that we earlier introduced the -category Dperf (R) whose objects
can be identied with bounded complexes of nite projective R-modules.
Denition 1. An inv
Introduction (Lecture 1)
February 2, 2011
In this course, we will be concerned with variations on the following:
Question 1. Let X be a CW complex. When does there exist a homotopy equivalence X
is a compact smooth manifold?
M , where M
In other words, wh