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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2011

    started adding some basic technical details to topological K-theory

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 31st 2012

    At topological K-theory I created a new subsection Relation to algebraic K-theory.

    So far it just points to two references, though.

    • CommentRowNumber3.
    • CommentAuthorTim_Porter
    • CommentTimeJan 31st 2012

    By the way, there is a strange \ref{} on the topological K-theory entry. I do not see what it should have been.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 31st 2012
    • (edited Jan 31st 2012)

    Thanks, I have fixed that.

    The problem was that the code for the definition had been

    +-- {: .num_defn#DefinitionOfKClasses}

    instead of

    +-- {: .num_defn #DefinitionOfKClasses}
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2012

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

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 13th 2016
    • (edited Dec 13th 2016)

    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.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2017
    • (edited May 26th 2017)

    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)K(X) and K˜(X)\tilde K(X) via virtual topological vector bundles, as well as their relation.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2017

    I have been adding some more of the basic stuff at

    But the entry does still remain incomplete.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2017
    • (edited May 30th 2017)

    I have been adding this and that to topological K-theory

    1. added definition of the graded K-groups (here)

    2. streamlined the long exact sequences (here)

    3. gave the external product on reduced groups its own subsection (here)

    4. added bare defintion of the product on graded K-groups (here)

    Not proof-read yet. Need to dash.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2017

    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”.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 6th 2017
    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2017

    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.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2017
    • (edited Jul 19th 2017)

    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

    1. starting with the proof of the Tietze extension theorem

    2. via the lemmas about extending bundle isomorphisms here

    3. to the corollary of the exactness here.

  1. Corrected typo at definition of multiplication.


    diff, v96, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeDec 12th 2018

    Thanks for catching!

  2. typos in ###Graded-commutative ring structre oringin: XY=(X×X)/(XY)X \wedge Y = (X \times X)/(X \vee Y)

    XΔ XX×XqX X \overset{\Delta_X}{\longrightarrow} X \times X \overset{q}{\longrightarrow} X

    change to: XX=(X×X)/(XX)X \wedge X = (X \times X)/(X \vee X)

    XΔ XX×XqXX X \overset{\Delta_X}{\longrightarrow} X \times X \overset{q}{\longrightarrow} X\vee X

    Joe Onekun

    diff, v98, current

  3. typo in ###Graded-commutative ring structure


    XΔ XX×XqXX X \overset{\Delta_X}{\longrightarrow} X \times X \overset{q}{\longrightarrow} X\vee X

    change to:

    XΔ XX×XqXX X \overset{\Delta_X}{\longrightarrow} X \times X \overset{q}{\longrightarrow} X\wedge X

    Joe Onekun

    diff, v98, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2020

    apparently the pdf link for

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

    diff, v114, current

    • CommentRowNumber19.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 5th 2020

    Uploading Wirthmüller’s lectures notes to the nLab.

    diff, v115, current

    • CommentRowNumber20.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 5th 2020

    Okay, I tried to upload the file, but I get

    500 Internal Server Error

    when trying to use the form for file upload at

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeDec 6th 2020

    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?

    • CommentRowNumber22.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 7th 2020

    Adjusting the file name…

    diff, v117, current

    • CommentRowNumber23.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 7th 2020

    Seems to work now, thanks.

  4. 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)


    diff, v127, current

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeJul 6th 2021

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

    diff, v128, current

  5. \tilde K to \widetilde{K}


    diff, v129, current

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2023

    added pointer to:

    • Raoul Bott, Lectures on K(X)K(X), Benjamin (1969) [pdf]

      Russian transl. by B. Yu. Sternin, Matematika 11 2 (1967) 32–56 [mathnet:mat424]

    diff, v131, current