This is a question on nomenclature of $K$-theory in the topological category.
The $K$-theory of a compact space $X$ is defined as the Grothendieck group of the vectorbundles on $X$. The Atiyah-Jänich Theorem states that this is the same thing as the homotopy classes of maps $X\rightarrow \Phi(\mathbb{H})$, where $\Phi(\mathbb{H})$ is the space of Fredholm operators on some separable infinite-dimensional Hilbert space.
Now, for a non-compact space $Y$ the $K$ theory $K(Y)$ is not defined as the Grothendieck group of vector bundles on $Y$. One can do a couple of things:
- Define it as the $K$-theory of its one point compactification
$K(Y):=K(Y_+)$. One needs to assume $Y$ is locally compact for this
to make sense. - Another option is to assume that $Y$ is nice, for example an
infinite $CW$ complex such as $CP^\infty$, and to define the $K$-theory of $Y$ as the limit of its finite subcomplexes. Note that $CP^\infty$ is not locally compact. - Yet another option is to define the $K$-theory of $Y$ as maps
$Y\rightarrow \Phi(\mathbb{H})$. I believe this is called representable
$K$-theory.
I have a couple of questions.
Why is 1. a good definition? I like my cohomology theories to be
functorial under maps. Theory 1. clearly is not. I believe it is functorial
with respect to proper maps. This reminds me of compactly supported
cohomology. But why is this then not called compactly supported $K$-theory?
And:
How are 2. and 3. related?
More specifically:
What exactly does 3. describe? Are these virtual vector bundles that admit numerable trivializations?
finally:
Is there a reference where all these definitions are discussed?
Best Answer
2 and 3 are not equivalent, because of a phenomenon known as "phantom maps".You can have a map of a CW complex $X$ to $Y$ which is non-trivial in homotopy, but homotopy trivial when restricted to every finite subcomplex of $X$.
That this actually occurs for K-cohomology is shown in an old paper of Anderson and Hodgkin, "the K-theory of Eilenberg Maclane Complexes" http://www.sciencedirect.com/science/article/pii/0040938368900098 See corollary 1 of that paper, which gives examples for Eilenberg-Maclane complexes.
Also, in answer to your question about an interpretation of 3 as virtual vector bundles, I don't think you will get any joy there. See the paper by Jackowski and Oliver quoted by Neil Strickland in his answer to this earlier MO question Is there a good definition of (topological) K-theory over arbitrary spaces