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 and and any two continuous maps , we say that is homotopic to is there is a continuous map such that , here the unit interval is given the Euclidean topology. In other words, is homotopic to if can vary continuously to . 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, . A map between two based spaces is a continuous map such that . 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 is a continuous map such that . Up to homotopy, a loop has an inverse, two loops can compose, and the constant map makes the set of loops in into a group, which we write as . We write for the space of loop spaces. This is a subspace of the map space , to which we give the compact open topology. This is again a based space with base point the constant map from to .
In order to introduce higher order homotopy groups, we need some operations in the category of (based) spaces. For two based spaces , we define their based union to be to be (with base point ), the is wedge product to be with base point . These two operations are really the analog of the disjoint union and Cartesian product in the category of topological spaces. We write for the wedge product of and where and the wedge product of and where . is called the cone of , and is called the suspension of . It is remarkable to see that , that is, the -sphere is the wedge product of an -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 is a map of based spaces . Note that this induces in an obvious way a map where is the spaces of maps of based spaces given the subspace topology induced from , the space of continuous maps from to , which is given the compact open topology. A co-multiplication on is a map . Again this induces a map (indeed, for any two , we can define as follows: for any , if takes to the first copy of in , then we set , otherwise . The importance of based spaces’ assumption is that, at the point , is well defined).
Are there some examples of multiplications and co-multiplications? In fact the loop space is a multiplication space, note that for any , the natural composition of gives the multiplication . The suspension is a co-multiplicaiton space, takes to if and to otherwise, where is the base point of .
For any with a co-multiplication or with multiplication, we see easily that has a monoid structure, yet in general this is not a group. We call a co--space if is a group for any based space , and we call an -space if is a group for any . We can verify that is an -space and is a co--space. At first sight, this is quite magical, at least this is so for me. Yet let’s recall how to show that is an -space. To construct the inverse of a loop in , we use essential the identification . In general, this is also the case. Recall that , a quotient space of , for any , we define . Now use the co-multiplication structure , and we write , then if , and otherwise. We need to show that is homotopic to the constant map . This is not hard to verify, just as in the case of . We omit the case that is an -space.
Now finally we can define the homotopy group. The -th homotopy group() of a based space is .
One surprising result is that is abelian for all . The usefulness of homotopy groups comes from Whitehead’s theorem:
For two CW-complexes , if the map induces isomorphisms for all , then 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 -theory.