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started adding some basic technical details to topological K-theory
At topological K-theory I created a new subsection Relation to algebraic K-theory.
So far it just points to two references, though.
By the way, there is a strange \ref{} on the topological K-theory entry. I do not see what it should have been.
Thanks, I have fixed that.
The problem was that the code for the definition had been
+-- {: .num_defn#DefinitionOfKClasses}
instead of
+-- {: .num_defn #DefinitionOfKClasses}
added to top. K-theory – classifying space statements about the model induced from unitaries modulo compact operators.
the entry topological K-theory was in a sad general state, and still is. But yesterday I tried to give it at least a half-way decent Idea-section.
I made this supercede the material that used to be in the Idea-section (which I felt at liberty to do, since probably I had written that some long time back).
I hope to come back to the entry and eventually turn it into something good. For the moment, due to lack of time, all I have to offer is an improved exposition in the Idea-section. But please feel invited to improve further.
I am starting to bring some more comprehensive details into topological K-theory.
Now I worked on the Definition section, spelling out in some detail the definition of the abelian groups $K(X)$ and $\tilde K(X)$ via virtual topological vector bundles, as well as their relation.
I have been adding some more of the basic stuff at
Bott periodicity (copied over from fundamental product theorem in topological K-theory)
But the entry does still remain incomplete.
I have been adding this and that to topological K-theory
added definition of the graded K-groups (here)
streamlined the long exact sequences (here)
gave the external product on reduced groups its own subsection (here)
added bare defintion of the product on graded K-groups (here)
Not proof-read yet. Need to dash.
I have added the remark (here) that one may think of the reduced K-groups of a compact Hausdorff spaces as those K-groups on any one-point complement space that “vanish at infinity”.
added to topological K-theory discussion of Complex orientation and Formal group law
I have completed writing up the explicit proof of the long exact sequences of topological K-theory groups by adding a few more lemmas and their proofs: here.
I have further polished some of the proofs in other entries that go into this. There is now detailed proof of the long exact sequences in topoogial K-theory over compact Hausdorff spaces
starting with the proof of the Tietze extension theorem
via the lemmas about extending bundle isomorphisms here
to the corollary of the exactness here.
Thanks for catching!
typos in ###Graded-commutative ring structre oringin: $X \wedge Y = (X \times X)/(X \vee Y)$
$X \overset{\Delta_X}{\longrightarrow} X \times X \overset{q}{\longrightarrow} X$change to: $X \wedge X = (X \times X)/(X \vee X)$
$X \overset{\Delta_X}{\longrightarrow} X \times X \overset{q}{\longrightarrow} X\vee X$Joe Onekun
apparently the pdf link for
is dead, and WaybackMachine doesn’t have a copy. Does anyone else? These were good lecture notes.
Okay, I tried to upload the file, but I get
500 Internal Server Error
when trying to use the form for file upload at
https://ncatlab.org/nlab/files/wirthm%C3%BCller-vector-bundles-and-k-theory.pdf
Thanks for looking into this!
I don’t know what the problem is, but I would try removing the special characters from the file name.
(If you replace the Umlaut by “ue” it’s still considered correct spelling.)
If this doesn’t work, maybe you could send me the file by email?
Seems to work now, thanks.
added pointer to:
Raoul Bott, Lectures on $K(X)$, Benjamin (1969) [pdf]
Russian transl. by B. Yu. Sternin, Matematika 11 2 (1967) 32–56 [mathnet:mat424]
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