notes on topological K-theory

This post is about topological K-theory. As for general information on this branch of mathematics, let’s quote some phrases from wikipedia: K-theory, roughly speaking, is the study of certain invariants of large matrices. In this post we will touch a very little bit of this K-theory. The materials of this post are taken from Husemoller’s book, ‘Fibre Bundles'(GTM82) and Hatcher’s bool, ‘algebraic topology’.


Here we list some foundational concepts for understanding K-theory.

Homotopy is the most basic one among these concepts. For any two topological spaces X and Y and any two continuous maps f,g:X\rightarrow Y, we say that f is homotopic to g is there is a continuous map F:X\times[0,1]\rightarrow Y such that F(x, 0)=f(x),F(x,1)=g(x),\forall x\in X, here the unit interval [0,1] is given the Euclidean topology. In other words, f is homotopic to g if f can vary continuously to g. It is not difficult to verify that homotopy is an equivalence relation. One major concern of algebraic topology is to classify topological spaces up to homotopy. The category of topological spaces with maps up to homotopy equivalence is a quotient category of the category of topological spaces with continuous maps.

Sometimes we study also based spaces and maps. A based space is a topological space with a point in it, (X,x_0). A map between two based spaces (X,x_0),(Y,y_0) is a continuous map f,X\rightarrow Y such that f(x_0)=y_0. The category of based topological spaces with maps up to homotopy equivalence is a quotient category of the topological spaces with continuous maps.

Next let’s define homotopy groups. A loop in a based topological space (X,x_0) is a continuous map f:[0,1]\rightarrow (X,x_0) such that f(0)=f(1)=x_0. Up to homotopy, a loop has an inverse, two loops can compose, and the constant map makes the set of loops in (X,x_0) into a group, which we write as \pi_1(X,x_0). We write \Omega(X) for the space of loop spaces. This is a subspace of the map space Map(I,X), to which we give the compact open topology. This is again a based space with base point the constant map from I to (X,x_0).

In order to introduce higher order homotopy groups, we need some operations in the category of (based) spaces. For two based spaces (X,x_0),(Y,y_0), we define their based union to be X\vee Y to be X\sqcup Y/(x_0\sim y_0)(with base point x_0=y_0), the is wedge product to be X\wedge Y=X\times Y/((x,y_0)\sim(x_0,y_0)\sim(x_0,y)) with base point (x_0,y_0). These two operations are really the analog of the disjoint union and Cartesian product in the category of topological spaces. We write C(X) for the wedge product of (X,x_0) and (I,0) where I=[0,1] and S(X) the wedge product of (X,x_0) and (S^1,0) where S^1=I/(0\sim1). C(X) is called the cone of X, and S(X) is called the suspension of X. It is remarkable to see that S^{n+1}=S(S^n), that is, the n+1-sphere is the wedge product of an n-sphere. It is not hard to imagine this identification, yet it takes some effort to write it down.

Next we say that a multiplication on a based space (Y,y_0) is a map of based spaces f:Y\times Y\rightarrow Y. Note that this f induces in an obvious way a map f_X: Map_0(X,Y)\times Map_0(X,Y)\rightarrow Map_0(X,Y) where Map_0(X,Y) is the spaces of maps of based spaces (X,x_0),(Y,y_0) given the subspace topology induced from Map(X,Y), the space of continuous maps from X to Y, which is given the compact open topology. A co-multiplication on (X,x_0) is a map g:X\rightarrow X\vee X. Again this g induces a map g^X:Map_0(X,Y)\times Map_0(X,Y)\rightarrow Map_0(X,Y)(indeed, for any two i,j:(X,x_0)\rightarrow(Y,y_0), we can define g^Y(i,j)\in Map_0(X,Y) as follows: for any x\in X, if g takes x to the first copy of X in X\vee X, then we set g^Y(i,j)(x)=i(x), otherwise g^Y(i,j)(x)=j(x). The importance of based spaces’ assumption is that, at the point x_0, g^Y(i,j) is well defined).

Are there some examples of multiplications and co-multiplications? In fact the loop space \Omega(Y) is a multiplication space, note that for any \alpha,\beta\in \Omega(Y), the natural composition of \alpha,\beta gives the multiplication f: \Omega(Y)\times \Omega(Y)\rightarrow \Omega(Y), (\alpha,\beta)\mapsto \alpha\circ\beta. The suspension S(X) is a co-multiplicaiton space, g:S(X)\rightarrow S(X)\vee S(X) takes (x,t)\in S(X)=X\wedge S^1 to ((x,2t),*) if 0\leq t\leq 1/2 and to (*,(x,2t-1)) otherwise, where * is the base point of S(X).

For any (X,x_0) with a co-multiplication or (Y,y_0) with multiplication, we see easily that Map_0(X,Y) has a monoid structure, yet in general this is not a group. We call (X,x_0) a co-H-space if Map_0(X,Y) is a group for any based space (Y,y_0), and we call (Y,y_0) an H-space if Map_0(X,Y) is a group for any (X,x_0). We can verify that S(X) is an H-space and \Omega(Y) is a co-H-space. At first sight, this is quite magical, at least this is so for me. Yet let’s recall how to show that S^1 is an H-space. To construct the inverse of a loop in (X,x_0), we use essential the identification S^1=I/\partial I. In general, this is also the case. Recall that S(X)=X\wedge S^1, a quotient space of X\times I, for any \alpha\in Map_0(S(X),Y), we define \alpha'(x,t)=\alpha(x,1-t)\in Y. Now use the co-multiplication structure g:S(X)\rightarrow S(X)\vee S(X), and we write g^Y(\alpha,\alpha')=\beta, then \beta(x,t)=\alpha(x,2t) if 0\leq t\leq 1/2, and \beta(x,t)=\alpha'(x,2t-1)=\alpha(x,2-2t) otherwise. We need to show that \beta is homotopic to the constant map const: S(X)\rightarrow Y,(x,t)\mapsto y_0. This is not hard to verify, just as in the case of S^1. We omit the case that \Omega(Y) is an H-space.

Now finally we can define the homotopy group. The n-th homotopy group(n>0) of a based space (X,x_0) is \pi_n(X,x_0)=Map_0(S^n,X).

One surprising result is that \pi_n(X,x_0) is abelian for all n>1. The usefulness of homotopy groups comes from Whitehead’s theorem:

For two CW-complexes X, Y, if the map f:X\rightarrow Y induces isomorphisms f_*: \pi_n(X)\rightarrow \pi_n(Y) for all n, then f is a homotopy equivalence.

So this theorem justifies the importance of the homotopy groups, yet they are very hard to compute.

In the next post we shall say something about fibre bundles, which turns out to be a fundamental tool in the development of K-theory.


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