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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 7th 2015
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   Author: Urs
   Format: MarkdownItexgave _[[Atiyah-Hirzebruch spectral sequence]]_ a minimum of an Idea-section and added a minimum paragraph with pointers to applications to D-brane charges in string theory [here](http://ncatlab.org/nlab/show/Atiyah–Hirzebruch+spectral+sequence#ApplicationDBraneChargesInStringTheory), also on the D-brane charge page itself [here](http://ncatlab.org/nlab/show/D-brane#DBraneChargeViaAtiyahHirzebruchSpectralSequence)

   gave Atiyah-Hirzebruch spectral sequence a minimum of an Idea-section and added a minimum paragraph with pointers to applications to D-brane charges in string theory here, also on the D-brane charge page itself here

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 10th 2016
   • (edited Feb 10th 2016)
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   Author: Urs
   Format: MarkdownItexAdded a _[note on terminology](https://ncatlab.org/nlab/show/Atiyah–Hirzebruch+spectral+sequence#NoteOnTerminology)_: > Often the terminology "Atiyah-Hirzebruch spectral sequence" is taken to refer to only this case with $F = \ast$, while the general case is then referred to as "Serre spectral sequence for generalized cohomology" or similar. In ([Atiyah-Hirzebruch 61,p. 17](#AtiyahHirzebruch61)) the case $F = \ast$ is labeled "Theorem", while the general case, stated right after the theorem, is labeled "2.2 Remark". > The proof of the theorem that is given is very short, it just says that since topological K-theory satisfies the [exactness axiom](generalized+%28Eilenberg-Steenrod%29+cohomology#ExactnessUnreduced) of a generalized cohomology theory, it is immediate that the conditions for a spectral sequence stated as Axioms (SP.1)-(SP.5) in ([Cartan-Eilenberg 56, section XV.7](#CartanEilenberg56)) are met. Indeed Example 2 in ([Cartan-Eilenberg 56, section XV.7](#CartanEilenberg56)) observes that the spectral sequence in question exists for $E$ "some fixed [[cohomology theory]]" because "Axioms (SP.1)-(SP.4) are consequences of usual properties of cohomology groups". > In view of this, the contribution of ([Atiyah-Hirzebruch 61](#AtiyahHirzebruch61)) would not be so much the observation of what is now called the AHSS, rather than the proof that K-theory indeed satisfies the axioms of a generalized cohomology theory. Indeed, according to ([Adams 74, p. 127-128, 215](#Adams74)), the AHSS was earlier observed by [[George Whitehead]] and "then became a folk-theorem" which was "eventually published by Atiyah and Hirzebruch". > Maybe it should be called the "Cartan-Eilenberg-Whitehead spectral sequence".

   Added a note on terminology:

   Often the terminology “Atiyah-Hirzebruch spectral sequence” is taken to refer to only this case with $F = \ast$, while the general case is then referred to as “Serre spectral sequence for generalized cohomology” or similar. In (Atiyah-Hirzebruch 61,p. 17) the case $F = \ast$ is labeled “Theorem”, while the general case, stated right after the theorem, is labeled “2.2 Remark”.

   The proof of the theorem that is given is very short, it just says that since topological K-theory satisfies the exactness axiom of a generalized cohomology theory, it is immediate that the conditions for a spectral sequence stated as Axioms (SP.1)-(SP.5) in (Cartan-Eilenberg 56, section XV.7) are met. Indeed Example 2 in (Cartan-Eilenberg 56, section XV.7) observes that the spectral sequence in question exists for $E$ “some fixed cohomology theory” because “Axioms (SP.1)-(SP.4) are consequences of usual properties of cohomology groups”.

   In view of this, the contribution of (Atiyah-Hirzebruch 61) would not be so much the observation of what is now called the AHSS, rather than the proof that K-theory indeed satisfies the axioms of a generalized cohomology theory. Indeed, according to (Adams 74, p. 127-128, 215), the AHSS was earlier observed by George Whitehead and “then became a folk-theorem” which was “eventually published by Atiyah and Hirzebruch”.

   Maybe it should be called the “Cartan-Eilenberg-Whitehead spectral sequence”.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 11th 2016
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   Author: Urs
   Format: MarkdownItexStarted spelling out the [proof](https://ncatlab.org/nlab/show/Atiyah%E2%80%93Hirzebruch+spectral+sequence#ProofOfAHSS) (of the existence and second page of the AHSS).

   Started spelling out the proof (of the existence and second page of the AHSS).

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 19th 2016
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   Author: Urs
   Format: MarkdownItexIn the proof of the [[Atiyah-Hirzebruch spectral sequence]] I have fixed the statement that the $E_1$-page is the cochain complex for singular cohomology to the statement that it is the cochain complex for [[cellular cohomology]]. Of course the cohomology groups coincide (which led to the previous glitch), but at this point of the proof the cochain complex itself matters. I have also spelled out the proof in more detail, including the actual argument that $d_1$ is the cellular coboundary operator. But for the moment I did this in another entry ([here](https://ncatlab.org/nlab/show/Introduction+to+Stable+homotopy+theory+--+S#AHSSExistence)) and added just a pointer to that from the previous page. (Eventually, once my edits have stabilized a bit more, I'll port it over. )

   In the proof of the Atiyah-Hirzebruch spectral sequence I have fixed the statement that the $E_1$-page is the cochain complex for singular cohomology to the statement that it is the cochain complex for cellular cohomology. Of course the cohomology groups coincide (which led to the previous glitch), but at this point of the proof the cochain complex itself matters.

   I have also spelled out the proof in more detail, including the actual argument that $d_1$ is the cellular coboundary operator. But for the moment I did this in another entry (here) and added just a pointer to that from the previous page.

   (Eventually, once my edits have stabilized a bit more, I’ll port it over. )

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 29th 2016
   • (edited Apr 29th 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have filled in text much expanding (with respect to the previous version) * the [statement](https://ncatlab.org/nlab/show/Atiyah%E2%80%93Hirzebruch+spectral+sequence#Statement) * the [traditional construction](https://ncatlab.org/nlab/show/Atiyah%E2%80%93Hirzebruch+spectral+sequence#ConstructionByFilteringTheBaseSpace) by filtering the base space Then I added a section on * the [alternative construction ](https://ncatlab.org/nlab/show/Atiyah%E2%80%93Hirzebruch+spectral+sequence#ConstructionByFilteringTheCoefficientSpectrum) by filtering the coefficient spectrum, because I finally found (thanks to Hisham Sati!) an article that proves that the two constructions indeed give isomorphic spectral sequences: It's ancient: [Maunder 63](https://ncatlab.org/nlab/show/Atiyah%E2%80%93Hirzebruch+spectral+sequence#Maunder63). I haven't checked in full detail, but it seems clear that what Maunder checks is isomorphic to the traditional AHSS is just what Mike observed follows neatly from a perspective of HoTT (in his writeup at [[homotopytypetheory:spectral sequences]])

   I have filled in text much expanding (with respect to the previous version)

   • the statement

   • the traditional construction by filtering the base space

   Then I added a section on

   • the alternative construction by filtering the coefficient spectrum,

   because I finally found (thanks to Hisham Sati!) an article that proves that the two constructions indeed give isomorphic spectral sequences:

   It’s ancient: Maunder 63.

   I haven’t checked in full detail, but it seems clear that what Maunder checks is isomorphic to the traditional AHSS is just what Mike observed follows neatly from a perspective of HoTT (in his writeup at spectral sequences (homotopytypetheory))