Algebraic Topology: Lecture 12

By popular request, the final four lectures of the course give an introduction the Calculus of Functors of Goodwillie. The main references are the papers:

T. Goodwillie: Calculus I: The first derivative of pseudoisotopy theory K-Theory 4 (1990), 1-27.

T. Goodwillie: Calculus II: Analytic functors, K-Theory 5 (1991), 295-332.

Here is a summary of todays lecture. A commutative diagram of spaces 

   X0 &rarr X1
   &darr     &darr
   X2 &rarr X12
is said to be k-cartesian if the canonical map 
   a = a(X) : X0 &rarr holim(X1 &rarr X12 &larr X2)
from the upper left-hand corner to the homotopy pull-back of the remaining part of the diagram is a k-connected map. The diagram is said to be k-cocartesian if the canonical map 
   b = b(X) : hocolim(X1 &larr X0 &rarr X2) &rarr X12
from the homotopy push-out of the diagram with the lower right-hand term deleted to the lower right-hand term is a k-connected map. The diagram is homotopy cartesian if it is k-cartesian for all k and homotopy cocartesian if it is k-cocartesian for all k. Now let 
   F : U &rarr T
be a functor from the category of spaces to the category of pointed spaces. We assume that F takes weak equivalences to weak equivalences. We say that F is excisive if for every homotopy cocartesian square 
   X0 &rarr X1
   &darr     &darr
   X2 &rarr X12
the induced square 
   F(X0) &rarr F(X1)
     &darr        &darr
   F(X2) &rarr F(X12)
is homotopy cartesian. If this is the case then the homotopy groups 
   Hq(X;F) = &piq(F(X))
define a generalized homology theory.

Let F : U &rarr T be a functor that takes weak equivalences to weak equivalences. We say that F is stably excisive if it satisfies the following hypothesis E(c,&kappa) for some integers c and &kappa: Given a homotopy cocartesian square 

   X0 &rarr X1
   &darr     &darr
   X2 &rarr X12
in which the maps X0 &rarr X1 and X0 &rarr X2 are k1-connected and k2-connected with k1 and k2 greater than or equal to &kappa, the induced square 
   F(X0) &rarr F(X1)
     &darr        &darr
   F(X2) &rarr F(X12)
is (k1+k1-c)-cartesian. (The notion of &rho -analyticity for the functor F is a similar condition that involves cubical diagrams of all dimensions. We will define &rho -analyticity in the following lectures.) Let UX be the category of spaces over X. An object is a map of spaces f : Y &rarr X, and a morphism from f : Y &rarr X to f' : Y' &rarr X is a map g : Y &rarr Y' such that f = gf'. There forgetful functor t : UX &rarr U that takes f : Y &rarr X to Y. We say that a square diagram in UX is homotopy cocartesian if the image by t is a homotopy cocartesian square in U. We then define what it means for a functor 
   &Phi : UX &rarr T
to be excisive and stably excisive as before. Assume that &Phi(g) is a weak equivalence whenever t(g) is a weak equivalence and suppose that &Phi satisfies hypothesis E(c,&kappa). Given f : Y &rarr X, we define a homotopy cocartesian square diagram 
    Y  &rarr CXY
    &darr      &darr
   CXY &rarr SXY
as follows: The space CXY is called the fiber-wise unreduced cone or the mapping cylinder of f : Y &rarr X. It is defined to be the quotient space of the disjoint union of Y x [0,1] and X by the equivalence relation generated by the relation that identifies (y,0) and f(y). The structure map CXY &rarr X maps (y,t) to f(y) and x to x. The space SXY is called the fiber-wise unreduced suspension of f : Y &rarr X. It is defined to be the quotient space of the disjoint union of two copies of CXY by the equivalence relation that identifies (y,1) in one copy with (y,1) in the other copy. The induced square of pointed spaces 
    &Phi(Y)  &rarr &Phi(CXY)
      &darr         &darr
   &Phi(CXY) &rarr &Phi(SXY)
is need not be homotopy cartesian, but we get an induced map 
   &Phi(Y) &rarr holim(&Phi(CXY) &rarr &Phi(SXY) &larr &Phi(SXY)) =: T&Phi(Y).
The functor T&Phi satisfies hypothesis E(c-1,&kappa -1) because the fiber-wise unreduced suspension functor increases connectivity by 1. It follows that the functor P&Phi defined by 
   P&Phi(Y) = hocolim T(n)&Phi(Y)
               n
is excisive. If F : U &rarr T is a stably excisive functor, we let &Phi = Ft and define 
   PXF(Y) = P&Phi(Y)
   DXF(Y) = homotopy fiber(PXF(Y) &rarr PXF(X)).
Then PXF : UX &rarr T is an excisive functor that we call the 1-jet of F at X and DXF : UX &rarr T is a linear functor that we call the differential of F at X.