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]

\tilde K to \widetilde{K}

Anonymous

]]>Thanks for catching this, that was a bad typo. In fact, also the text around it was wrong. I have fixed it now.

]]>Corrected typo in the definition of stably equivalent vector bundles; It should read E \oplus (X \times k^n) instead of E \otimes (X \times k^n)

Anonymous

]]>Seems to work now, thanks.

]]>Adjusting the file name…

]]>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?

]]>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

]]>Uploading Wirthmüller’s lectures notes to the nLab.

]]>apparently the pdf link for

- Klaus Wirthmüller,
*Vector bundles and K-theory*, 2012 (pdf)

is dead, and WaybackMachine doesn’t have a copy. Does anyone else? These were good lecture notes.

]]>typo in ###Graded-commutative ring structure

oringin:

$X \overset{\Delta_X}{\longrightarrow} X \times X \overset{q}{\longrightarrow} X\vee X$change to:

$X \overset{\Delta_X}{\longrightarrow} X \times X \overset{q}{\longrightarrow} X\wedge X$Joe Onekun

]]>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

]]>Thanks for catching!

]]>Corrected typo at definition of multiplication.

Anonymous

]]>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.

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.

]]>added to topological K-theory discussion of *Complex orientation and Formal group law*

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”.

]]>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 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 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.

]]>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.

]]>added to *top. K-theory – classifying space* statements about the model induced from unitaries modulo compact operators.

Thanks, I have fixed that.

The problem was that the code for the definition had been

```
+-- {: .num_defn#DefinitionOfKClasses}
```

instead of

```
+-- {: .num_defn #DefinitionOfKClasses}
```

]]>