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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 15th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded brief definition/characterization to [[Chern class]]

   added brief definition/characterization to Chern class

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 29th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded section _[Chern class -- Properties -- Splitting principle and Chern roots](http://ncatlab.org/nlab/show/Chern+class#SplittingPrinciple)_.

   added section Chern class – Properties – Splitting principle and Chern roots.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 22nd 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave spelled out a proof [here](https://ncatlab.org/nlab/show/Chern+class#GeneratorsOfCohomologyOfBunChernClasses), via induction over the Thom-Gysin sequence, of the basic fact $H^\bullet(B U(n))\simeq \mathbb{Z}[ c_1,\cdots , c_n ]$

   have spelled out a proof here, via induction over the Thom-Gysin sequence, of the basic fact $H^\bullet(B U(n))\simeq \mathbb{Z}[ c_1,\cdots , c_n ]$

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 24th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have spelled out the proof of the splitting principle for Chern classes [here](https://ncatlab.org/nlab/show/Chern+class#SplittingPrincipleForChernClasses) (modulo the lemma that pullback in cohomology along $B U(1)^n \to B U(n)$ is injective).

   I have spelled out the proof of the splitting principle for Chern classes here (modulo the lemma that pullback in cohomology along $B U(1)^n \to B U(n)$ is injective).

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 25th 2016
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexThe proof that it is injective could go at a page dealing with maximal tori, presumably? Should hold for all $BT \to BG$, for reasonable $G$...

   The proof that it is injective could go at a page dealing with maximal tori, presumably? Should hold for all $BT \to BG$, for reasonable $G$…

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 27th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexDo you have a pointer to a proof (in more generality or not)? That argument in Kochmann's book is a little shaky.

   Do you have a pointer to a proof (in more generality or not)? That argument in Kochmann’s book is a little shaky.

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 27th 2016
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI emailed you, but for others: Johannes Ebert, in [this MO answer](http://mathoverflow.net/a/61943/4177) argues (following Dupont) that $H^*(BG) \to H^*(BT)$ is, under the Chern-Weil isomorphism for compact (connected?) $G$, $Sym^{\ast} \mathfrak{g}^{\vee} \to Sym^{\ast} \mathfrak{t}^{\vee}$, and in fact just multiplication by $\chi(G/T)$. This Euler characteristic is non-zero by a Lefshetz fixed-point argument involving the action of $G$ on $G/T$.

   I emailed you, but for others: Johannes Ebert, in this MO answer argues (following Dupont) that $H^*(BG) \to H^*(BT)$ is, under the Chern-Weil isomorphism for compact (connected?) $G$, $Sym^{\ast} \mathfrak{g}^{\vee} \to Sym^{\ast} \mathfrak{t}^{\vee}$, and in fact just multiplication by $\chi(G/T)$. This Euler characteristic is non-zero by a Lefshetz fixed-point argument involving the action of $G$ on $G/T$.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJun 27th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. Do you have the energy left to make a note about this on some $n$Lab page? Best place might be _[[splitting principle]]_.

   Thanks. Do you have the energy left to make a note about this on some $n$Lab page? Best place might be splitting principle.

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 27th 2016
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexCan do.

   Can do.

10. • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 28th 2016
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexI've added a little something to [[splitting principle]], more just a record of the argument, with a citation of Dupont.

    I’ve added a little something to splitting principle, more just a record of the argument, with a citation of Dupont.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2016
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks! Here is a [pointer](https://ncatlab.org/nlab/show/splitting+principle#InjectivityOfPullbackInCohomologyToBT) to the remark that you added.

    Thanks! Here is a pointer to the remark that you added.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2016
    • PermaLink
    Author: Urs
    Format: MarkdownItexIn your [paragraph](https://ncatlab.org/nlab/show/splitting+principle#InjectivityOfPullbackInCohomologyToBT) I have made "transfer" point to _[[Becker-Gottlieb transfer]]_ and changed the pointer _[[Euler characteristic]]_ to _[[Euler class]]_.

    In your paragraph I have made “transfer” point to Becker-Gottlieb transfer and changed the pointer Euler characteristic to Euler class.

13. • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 28th 2016
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexHmm, I should point out at [[Euler class]] that the Euler characteristic is actually a number, not just a cohomology class.

    Hmm, I should point out at Euler class that the Euler characteristic is actually a number, not just a cohomology class.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2016
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have also added proof of the Whitney sum formula for Chern classes, [here](https://ncatlab.org/nlab/show/Chern+class#WhitneySumFormula).

    I have also added proof of the Whitney sum formula for Chern classes, here.

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 28th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to * [[Shoshichi Kobayashi]], [[Katsumi Nomizu]], Section XII.3 in: _Foundations of Differential Geometry, Volume 1_, Wiley 1963 ([web](https://www.zuj.edu.jo/download/foundations-of-differential-geometry-vol-1-kobayashi-nomizu-pdf/), [ISBN:9780471157335](https://www.wiley.com/en-us/Foundations+of+Differential+Geometry%2C+Volume+1-p-9780471157335), [Wikipedia](https://en.wikipedia.org/wiki/Foundations_of_Differential_Geometry)) <a href="https://ncatlab.org/nlab/revision/diff/Chern+class/37">diff</a>, <a href="https://ncatlab.org/nlab/revision/Chern+class/37">v37</a>, <a href="https://ncatlab.org/nlab/show/Chern+class">current</a>

    added pointer to

    • Shoshichi Kobayashi, Katsumi Nomizu, Section XII.3 in: Foundations of Differential Geometry, Volume 1, Wiley 1963 (web, ISBN:9780471157335, Wikipedia)

    diff, v37, current

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to * [[Friedrich Hirzebruch]], Chapter 1, Section 4 of: _Neue topologische Methoden in der Algebraischen Geometrie_, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge, Springer 1956 ([doi:10.1007/978-3-662-41083-7](https://www.springer.com/de/book/9783662406052)) <a href="https://ncatlab.org/nlab/revision/diff/Chern+class/38">diff</a>, <a href="https://ncatlab.org/nlab/revision/Chern+class/38">v38</a>, <a href="https://ncatlab.org/nlab/show/Chern+class">current</a>

    added pointer to

    • Friedrich Hirzebruch, Chapter 1, Section 4 of: Neue topologische Methoden in der Algebraischen Geometrie, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge, Springer 1956 (doi:10.1007/978-3-662-41083-7)

    diff, v38, current

17. • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 4th 2021
    • (edited Feb 4th 2021)
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexSomeone on Twitter was asking about the geometric meaning of Chern classes. I mentioned the nPOV on [[characteristic classes]]. Now, I get the idea that unitary group bundles over a space $X$ are mapped to ordinary cohomology: $[X, B U] \mapsto [X, B^{2i} \mathbb{Z}]$ by cocycle composition. But I hadn't thought what the latter is qua principal $B^{2i} \mathbb{Z}$-bundle. Is there a geometric picture of this, e.g., in the case $i = 1$, the $B^2 \mathbb{Z}$-bundle associated to a circle-bundle via $c_1$? I see that in the area of [enumerative geometry](https://fulges.github.io/docs/Teaching/2021_MPI_EnumerativeGeometry/IntroEnumerativeGeometry.pdf), I'd need to be thinking about Chow rings > Chern classes are elements of the Chow ring.

    Someone on Twitter was asking about the geometric meaning of Chern classes. I mentioned the nPOV on characteristic classes. Now, I get the idea that unitary group bundles over a space $X$ are mapped to ordinary cohomology: $[X, B U] \mapsto [X, B^{2i} \mathbb{Z}]$ by cocycle composition. But I hadn’t thought what the latter is qua principal $B^{2i} \mathbb{Z}$-bundle.

    Is there a geometric picture of this, e.g., in the case $i = 1$, the $B^2 \mathbb{Z}$-bundle associated to a circle-bundle via $c_1$?

    I see that in the area of enumerative geometry, I’d need to be thinking about Chow rings

    Chern classes are elements of the Chow ring.

18. • CommentRowNumber18.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 4th 2021
    • (edited Feb 4th 2021)
    • PermaLink
    Author: Richard Williamson
    Format: MarkdownItexMaybe it's not what you are looking for, but Chern classes are basically algebraic/geometrical versions of topological homology classes: i.e. instead of looking at topological subspaces, one looks at (a generalisation of) sub-varieties. This is why Chow groups are closely related to motivic phenomena: in some sense, motives are an attempt to build a 'new geometry' out of these algebraic analogues of homology classes, or more precisely to obtain/express results which hold in general for them. One can always 'forget' some information, and that is how one ends up with cycle maps, i.e. cohomological versions of Chern classes. Thus, for a smooth complex variety, we can 'forget' the geometry of the 'algebraic/geometrical homology classes' and just remember their topology, and this gives a cycle map to (topological) singular cohomology. Over a finite field, one uses étale, crystalline, etc cohomology instead. Of course forgetting information often leads to something more computable. One can forget 'a little less', and one then ends up with cycle maps into things like Deligne cohomology. In short, though, one is just looking at how a variety can be built up from sub-varieties of it (up to an algebraic analogue of homotopy, namely rational equivalence). Riemann-Roch says that one can equivalently speak of (algebraic) vector bundles.

    Maybe it’s not what you are looking for, but Chern classes are basically algebraic/geometrical versions of topological homology classes: i.e. instead of looking at topological subspaces, one looks at (a generalisation of) sub-varieties. This is why Chow groups are closely related to motivic phenomena: in some sense, motives are an attempt to build a ’new geometry’ out of these algebraic analogues of homology classes, or more precisely to obtain/express results which hold in general for them.

    One can always ’forget’ some information, and that is how one ends up with cycle maps, i.e. cohomological versions of Chern classes. Thus, for a smooth complex variety, we can ’forget’ the geometry of the ’algebraic/geometrical homology classes’ and just remember their topology, and this gives a cycle map to (topological) singular cohomology. Over a finite field, one uses étale, crystalline, etc cohomology instead. Of course forgetting information often leads to something more computable. One can forget ’a little less’, and one then ends up with cycle maps into things like Deligne cohomology.

    In short, though, one is just looking at how a variety can be built up from sub-varieties of it (up to an algebraic analogue of homotopy, namely rational equivalence). Riemann-Roch says that one can equivalently speak of (algebraic) vector bundles.

19. • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexNot sure if I understand yet properly what the question is after (is it about finding intuitively accessible geometric representatives of objects classified by the universal Chern classes?), but here are two quick comments: 1. at least the second universal Chern class on $B SU(2) \simeq B Sp(1)$ may be understood as being "like" the first is for complex line bundles, but now for quaternionic line bundles. This is the eventual content of the discussion (or monologue) on quaternionic orientation from complex orientation [here](https://nforum.ncatlab.org/discussion/11962/quaternionic-oriented-cohomology-theory/?Focus=89221#Comment_89221). 1. one bit of algebraic topology analogous to Chow groups is (not $BU$ but) [[MU]] (which is "close", of course, in a sense), complex cobordism: By the article and its MO commentary briefly recorded [there](https://ncatlab.org/nlab/show/MU#RelationToDivisors).

    Not sure if I understand yet properly what the question is after (is it about finding intuitively accessible geometric representatives of objects classified by the universal Chern classes?), but here are two quick comments:

    1. at least the second universal Chern class on $B SU(2) \simeq B Sp(1)$ may be understood as being “like” the first is for complex line bundles, but now for quaternionic line bundles. This is the eventual content of the discussion (or monologue) on quaternionic orientation from complex orientation here.

    2. one bit of algebraic topology analogous to Chow groups is (not $BU$ but) MU (which is “close”, of course, in a sense), complex cobordism: By the article and its MO commentary briefly recorded there.

20. • CommentRowNumber20.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 4th 2021
    • PermaLink
    Author: Richard Williamson
    Format: MarkdownItexYes, complex cobordism is definitely related: from the point of view of my comment, one is still forgetting something, but in a slightly different way. Hence there is for example a 'motivic cobordism theory' (i.e. a generalised motivic cohomology') which 'lifts' the algebro-topological cobordism theory into the motivic world.

    Yes, complex cobordism is definitely related: from the point of view of my comment, one is still forgetting something, but in a slightly different way. Hence there is for example a ’motivic cobordism theory’ (i.e. a generalised motivic cohomology’) which ’lifts’ the algebro-topological cobordism theory into the motivic world.

21. • CommentRowNumber21.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 4th 2021
    • PermaLink
    Author: Richard Williamson
    Format: MarkdownItexI can maybe add that the following... > But I hadn't thought what the latter is qua principal $B^{2i} \mathbb{Z}$-bundle. ...is basically the analogue in a algebraic/geometrical setting of finding 'nice' representatives of a homology class. This is a difficult/interesting problem in general, it is like asking for an analogue of the description of $K(\mathbb{Z}, 2)$ as $\mathbb{C}\mathbb{P}^{\infty}$. This is the kind of thing Urs was getting at in 1. in one case, I think.

    I can maybe add that the following…

    But I hadn’t thought what the latter is qua principal $B^{2i} \mathbb{Z}$-bundle.

    …is basically the analogue in a algebraic/geometrical setting of finding ’nice’ representatives of a homology class. This is a difficult/interesting problem in general, it is like asking for an analogue of the description of $K(\mathbb{Z}, 2)$ as $\mathbb{C}\mathbb{P}^{\infty}$. This is the kind of thing Urs was getting at in 1. in one case, I think.

22. • CommentRowNumber22.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 4th 2021
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThanks both! Plenty of useful ideas there.

    Thanks both! Plenty of useful ideas there.

23. • CommentRowNumber23.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 4th 2021
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexJust as a sanity check, the shift to universal Chern classes on $B SU(n)$ rather than $B U(n)$ in #19 point 1 is made because we care about $SU(2)$ rather than $U(2)$, and is just mediated by the inclusion of $SU(2)$ into $U(2)$?

    Just as a sanity check, the shift to universal Chern classes on $B SU(n)$ rather than $B U(n)$ in #19 point 1 is made because we care about $SU(2)$ rather than $U(2)$, and is just mediated by the inclusion of $SU(2)$ into $U(2)$?

24. • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexYes, I considered restricting $c_2$ along $SU(2) \overset{i}{\hookrightarrow} U(n)$, or rather along $B SU(2) \overset{B i}{\longrightarrow} B U(n)$ only to make use of the exceptional isomorphism $SU(2) \simeq Sp(1)$, since the latter group [[Sp(1)]] is for quaternionic line bundles what [[U(1)]] is for complex line bundles. By the way, not to forget that $c_2$ in general (i.e. without such restrictions) also "is" the universal [[Chern-Simons circle 3-bundle]].

    Yes, I considered restricting $c_2$ along $SU(2) \overset{i}{\hookrightarrow} U(n)$, or rather along $B SU(2) \overset{B i}{\longrightarrow} B U(n)$ only to make use of the exceptional isomorphism $SU(2) \simeq Sp(1)$, since the latter group Sp(1) is for quaternionic line bundles what U(1) is for complex line bundles.

    By the way, not to forget that $c_2$ in general (i.e. without such restrictions) also “is” the universal Chern-Simons circle 3-bundle.

25. • CommentRowNumber25.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 4th 2021
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThanks! And of course instantons make us care about $SU(2)$.

    Thanks! And of course instantons make us care about $SU(2)$.

26. • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexOh, we care about $c_2$ on $B SU(2)$ for many reasons, also for chiral perturbation theory and Skyrmions and heterotic line bundles and ... all appearing at once in the Hypothesis H M5-brane model (see on the right of the big diagram [here](https://ncatlab.org/schreiber/show/Twisted+Cohomotopy+implies+twisted+String+structure+on+M5-branes)).

    Oh, we care about $c_2$ on $B SU(2)$ for many reasons, also for chiral perturbation theory and Skyrmions and heterotic line bundles and … all appearing at once in the Hypothesis H M5-brane model (see on the right of the big diagram here).

27. • CommentRowNumber27.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 14th 2021
    • PermaLink
    Author: Dmitri Pavlov
    Format: MarkdownItexAdded: * Timothy Hosgood, _Chern classes of coherent analytic sheaves: a simplicial approach_. Université d’Aix-Marseille (AMU), 2020. [tel-02882140](https://tel.archives-ouvertes.fr/tel-02882140). A [[MathOverflow]] question discussing references: <https://mathoverflow.net/questions/345437/canonical-reference-for-chern-characteristic-classes> <a href="https://ncatlab.org/nlab/revision/diff/Chern+class/44">diff</a>, <a href="https://ncatlab.org/nlab/revision/Chern+class/44">v44</a>, <a href="https://ncatlab.org/nlab/show/Chern+class">current</a>

    Added:

    • Timothy Hosgood, Chern classes of coherent analytic sheaves: a simplicial approach. Université d’Aix-Marseille (AMU), 2020. tel-02882140.

    A MathOverflow question discussing references: https://mathoverflow.net/questions/345437/canonical-reference-for-chern-characteristic-classes

    diff, v44, current