Talk:Gysin homomorphism

Sphere bundle

It surely needs to be mentioned that the sphere bundle in question should be oriented, unless one is using twisted or Z/2 coefficients. 31.205.108.80 (talk) 14:01, 14 June 2013 (UTC)

Done. -- Taku (talk) 20:03, 19 July 2017 (UTC)

Merge proposal

On 05:49, 27 December 2014‎ User:TakuyaMurata proposed that Shriek map and this article Gysin sequence, be merged. He did not explain why. Below is space for a discussion.

• I'd just like to cast a vote AGAINST merging the shriek map page with this one as they are quite distinct concepts. The Gysin sequence is more fundamental to topology while the shriek is more fundamental to algebraic geometry.Skeesix (talk) 15:16, 8 March 2015 (UTC)

• Half-merge -- currently, transfer map redirects to Shriek map, where it is explained, as having to do with Gysin sequences and algebraic topology. How about merging only that subsection to here? That makes sense to me, and is consistent with the above remark... 67.198.37.16 (talk) 04:42, 5 September 2016 (UTC)

Add algebraic geometry section

I have started writing up a section for defining the gysin morphism...

Algebraic Geometry

The Gysin morphism

$i^{!}:A_{d}(Y)\to A_{d-r}(X)$

for a regular embedding $i:X\to Y$ of codimension $r$ where $X,Y$ are schemes of finite type over a field can be defined as follows: consider a dimension $d$ subvariety $W\hookrightarrow Y$. From the pullback square

${\begin{matrix}V&\to &W\\q\downarrow &&\downarrow \\X&{\xrightarrow {i}}&Y\end{matrix}}$

we can construct a bundle over $V$, the pullback of the normal bundle $\pi :q^{*}N_{X/Y}\to V$, and a variety $C_{V/W}$ embedding into this bundle. We define this variety as

$C_{V/W}:={\text{Spec}}\left({\text{Sym}}^{\bullet }\left({\frac {J}{J^{2}}}\right)\right):={\text{Spec}}\left(\bigoplus _{n\geq 0}{\frac {J^{n}}{J^{n+1}}}\right)$

where $J$ is the ideal sheaf $I_{V/W}$. Since we have an isomorphism $\pi ^{*}:A_{d}(V)\to A_{d+r}(q^{*}N_{X/Y})$ we can take

I also had a draft on the definition of Fulton's refined Gysin homomorphism. I just have put it to the article. Maybe you want to expand/rewrite it with your materials. -- Taku (talk) 08:14, 18 July 2017 (UTC)

Add Complex Geometry Sections

Given a complex submanifold $Y\hookrightarrow X$ there is a gysin morphism from the cohomology of $Y$ to the cohomology of $X$. This morphism is given by the top chern class of the normal bundle. This page should include this construction and give examples.

Add construction with spectral sequences

The Gysin and Wang morphisms can be constructed using spectral sequences. Check out these *very informative* notes https://www.math.wisc.edu/~maxim/spseq.pdf — Preceding unsigned comment added by Wundzer (talk • contribs) 01:57, 11 February 2020 (UTC)