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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 25th 2016
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   Author: Urs
   Format: MarkdownItexI have spelled out the derivation of the Gysin sequence from the Serre spectral sequence at _[[Gysin sequence]]_ .

   I have spelled out the derivation of the Gysin sequence from the Serre spectral sequence at Gysin sequence .

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 3rd 2016
   • (edited Jun 3rd 2016)
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   Author: Urs
   Format: MarkdownItexI have expanded the proof at _[[Gysin sequence]]_ a tad, including some pedantery in an attempt to make it crystal clear why the steps hold. Most accounts gloss over some subtleties. For instance for concluding in the proof that every element in $E_2^{0,n}$ is of the form $\iota \cdot b$ with $\iota$ the unit in $H^\bullet(B;R)$ but regarded in $E_2^{0,n}$, this really needs an argument that the product $$ E_2^{\bullet,0} \otimes E_2^{\bullet,n} \longrightarrow E_2^{0,n} $$ in the multiplicative spectral sequence here is indeed just the cup product in the cohomology ring, under the identification $$ E_2^{\bullet,0} \simeq E_2^{\bullet,n} \simeq H^\bullet(B;R) \,. $$ This needs inspection of the definition of the multiplicative structure, which is tedious (as witnessed by the discussion at _[[multiplicative spectral sequence]]_) and rarely spelled out. I have added pointer to [Kochman 96, first equation in the proof of prop. 4.2.9](https://ncatlab.org/nlab/show/Thom-Gysin+sequence#Kochmann96), from which one may see it. I got into this when trying to write out an elegant proof of the Thom isomorphism not via "relative Serre spectral sequences" but via just applyng the Gysin sequence to the _fiberwise_ cofiber of $S(V) \to D(V)$. I think that works, but it does require fine control over how these steps in the Gysin sequence work. Or so it seems to me.

   I have expanded the proof at Gysin sequence a tad, including some pedantery in an attempt to make it crystal clear why the steps hold.

   Most accounts gloss over some subtleties. For instance for concluding in the proof that every element in $E_2^{0,n}$ is of the form $\iota \cdot b$ with $\iota$ the unit in $H^\bullet(B;R)$ but regarded in $E_2^{0,n}$, this really needs an argument that the product

   $$E_2^{\bullet,0} \otimes E_2^{\bullet,n} \longrightarrow E_2^{0,n}$$

   in the multiplicative spectral sequence here is indeed just the cup product in the cohomology ring, under the identification

   $$E_2^{\bullet,0} \simeq E_2^{\bullet,n} \simeq H^\bullet(B;R) \,.$$

   This needs inspection of the definition of the multiplicative structure, which is tedious (as witnessed by the discussion at multiplicative spectral sequence) and rarely spelled out. I have added pointer to Kochman 96, first equation in the proof of prop. 4.2.9, from which one may see it.

   I got into this when trying to write out an elegant proof of the Thom isomorphism not via “relative Serre spectral sequences” but via just applyng the Gysin sequence to the fiberwise cofiber of $S(V) \to D(V)$. I think that works, but it does require fine control over how these steps in the Gysin sequence work. Or so it seems to me.

3. • CommentRowNumber3.
   • CommentAuthorUlrik
   • CommentTimeJun 3rd 2016
   • PermaLink
   Author: Ulrik
   Format: MarkdownItexBTW, Guillaume Brunerie has worked out the Gysin sequence and the Thom isomorphism in HoTT in his recent thesis (which I don't think is available yet). He talked about it in Bonn and recently at the Fields Institute in Toronto: [abstract and video](http://www.fields.utoronto.ca/talks/fourth-homotopy-group-three-dimensional-sphere).

   BTW, Guillaume Brunerie has worked out the Gysin sequence and the Thom isomorphism in HoTT in his recent thesis (which I don’t think is available yet). He talked about it in Bonn and recently at the Fields Institute in Toronto: abstract and video.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 3rd 2016
   • (edited Jun 3rd 2016)
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   Author: Urs
   Format: MarkdownItexThanks for the information. Do you know at which minutes in the video these are mentioned? (I tried to look for it, but the board is unreadable in the recording, so I would have to listen to the whole thing, which I don't have the time for.)

   Thanks for the information. Do you know at which minutes in the video these are mentioned?

   (I tried to look for it, but the board is unreadable in the recording, so I would have to listen to the whole thing, which I don’t have the time for.)

5. • CommentRowNumber5.
   • CommentAuthorUlrik
   • CommentTimeJun 3rd 2016
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   Author: Ulrik
   Format: MarkdownItexProbably towards the end, but I don't remember the minute marks, sorry.

   Probably towards the end, but I don’t remember the minute marks, sorry.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 3rd 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay, thanks. Strange that the Fields institute doesn't produce usable video recordings. Somebody should contact them about this...

   Okay, thanks. Strange that the Fields institute doesn’t produce usable video recordings. Somebody should contact them about this…

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJun 21st 2016
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   Author: Urs
   Format: MarkdownItexI have now added pointer to [Brunerie 16](https://ncatlab.org/nlab/show/Thom-Gysin+sequence#Brunerie16)

   I have now added pointer to Brunerie 16