chemistry

#Contents#
* table of contents
{:toc}

## Idea

Chemistry is the branch of physical science that studies the properties of [[condensed matter]], from the scale of [[atom (physics)|atoms]] upwards.

A central part of chemistry is the identification of reaction processes -- _chemical reactions_ -- by which a variety of chemical substances (species of [[atom (physics)|atoms]] or [[molecules]]) interact to form new combinations. Such processes are typically denoted by formulas of the form

$$
  a A + b B \leftrightarrow c C + d D
$$

where the capital letters denote species of [[atom (physics)|atoms]]/[[molecules]], while the lower case letters are [[natural numbers]]. This indicates that $a$ molecules of type $A$ may react with $b$ molecules of type $B$ to $c$ molecules of type $C$ together with $d$ molecules of type $D$.

In [[thermodynamic equilibrium]] such reactions occur in both directions, whence the double arrow. A reaction of the form

$$
  A \to c C + d D
$$

would be called a _decay_ process. Notice that analogous processes appear elsewhere, for instance there are [[nuclear reactions]] in [[particle physics]] described by the same kind of reaction diagrams. Also [[D-branes]] for the [[topological string]] undergo reaction processes of this form, see [below](#TriangulatedCategoryWithStability).

One might be tempted to recognize [[morphisms]] as in [[category theory]] in this notation, but care needs to be exercised to arrive at a sensible concept.

## Formalization of reaction processes

It is unlikely that any formalization of reaction processes (say in [[category theory]]) will depend on the specific nature of the [[chemical elements]]. Among formalizations that do model reaction processes akin to those seen in chemistry are _stochastic Petri nets_ and _triangulated categories with stability conditions_:

### Stochastic Petri nets or Chemical reaction networks

Given a set $S$ of 'species', a **complex** of those species is a function $C : S \to \mathbb{N}$. A **reaction network** is a set of species together with a directed multigraph whose vertices are labelled by complexes of those species. Edges of the multigraph correspond to transitions.

Any reaction network gives rise to a [[Petri net]], and _vice versa_. In either case, we can label each transition by a rate constant in $(0, \infty)$. This gives **stochastic** versions of the nets and networks. (See [BaezNet](#BaezNet).)

### Triangulated categories with stability conditions
 {#TriangulatedCategoryWithStability}

One successful formalization in [[category theory]] of the idea of reaction processes appears in the context of [[D-branes]] for the [[B-model]] [[topological string]]. These are (hypothetical) physical objects that appear in different species labeled by [[objects]] in a [[triangulated category|triangulated]] [[derived category]] (of [[quasicoherent sheaves]] on some [[Calabi-Yau variety]]). There are processes

$$
  A \leftrightarrow B \oplus C
$$

by which a brane of type $A$ may decay into a brane of type B and a brane of type C.

While these are not reactions of [[chemical elements]], clearly the general form of reaction processes is similar.

And for the case of these D-branes, there happens to be a well-established and useful mathematical formalization of these reaction processes via [[category theory]]:

Namely every reaction process as above corresponds to a [[homotopy fiber sequence]] (a "distinguished triangle") in the [[triangulated category]], of the form

$$
  B \longrightarrow A \longrightarrow C
  \,.
$$

(See [Aspinwall 04](#Aspinwall04), search the document for the keyword "decay".)

Mathematically such a [[homotopy fiber sequence]] in a [[triangulated category]] precisely expresses the fact that $A$ is a "twisted [[direct sum]]" of $B$ and $C$ ([[extension]], [[semidirect product]]), hence much like the plain [[direct sum]], but with some "interaction" included.

In addition there are _[[Bridgeland stability conditions]]_ on such [[triangulated categories]] of topological D-branes. These determine which of these reaction processes lead to stable compounds, i.e. whether, in the above example, the brane of type $A$ will really decay into branes of type B and C, or if conversely the latter will fuse. (See again [Aspinwall 04](#Aspinwall04), search the document for the keyword "stability".)

After its motivation from [[D-brane]] reaction processes, the study of [[triangulated categories]] with [[Bridgeland stability conditions]] has become a rich and active area of pure [[mathematics]] in itself.

## Related $n$Lab entries

* [[quantum chemistry]]

* [[physics]]

## References

* Wikipedia, _[Chemistry](http://en.wikipedia.org/wiki/Chemistry)_

* Wikipedia, _[Chemical reaction](https://en.wikipedia.org/wiki/Chemical_reaction)_

* {#BaezNet} [[John Baez]], [chemical reaction networks](http://math.ucr.edu/home/baez/networks/)

On chemical [[topological phases of matter]] ("topological materials"):

* Bradlyn et al. *Topological quantum chemistry*, Nature **547** (2017) 298–305 [[doi:10.1038/nature23268](https://doi.org/10.1038/nature23268)]

On some kind of "chemistry" of [[D-brane]] reactions in [[topological string theory]]:

* {#Aspinwall04} [[Paul Aspinwall]], _D-Branes on Calabi-Yau Manifolds_ ([arXiv:hep-th/0403166](https://arxiv.org/abs/hep-th/0403166))

[[!redirects chemical reaction]]
[[!redirects chemical reactions]]


Chen connection

## Related concepts

* [[connection on a bundle]]

* [[iterated integral]]

## References

* [[Kuo Tsai Chen]], _Smooth maps, pullback path spaces, connections, and torsions_, Trans. Amer. Math. Soc. 297 (1986), 617-627 ([web](http://www.ams.org/journals/tran/1986-297-02/S0002-9947-1986-0854088-1/))

[[!redirects Chen connections]]


Chen Ning Yang

* [Wikipedia entry](http://en.wikipedia.org/wiki/Chen_Ning_Yang)

## Selected writings

Introducing [[Yang-Mills theory]] (eventually: [[quantum chromodynamics]]):

* [[Chen Ning Yang]], [[Robert Mills]], _Conservation of Isotopic Spin and Isotopic Gauge Invariance_. Physical Review 96 (1): 191– 195. (1954) ([web](http://prola.aps.org/abstract/PR/v96/i1/p191_1))

On the [[braid group]], [[knot theory]] and [[braid representations]] via the [[quantum Yang-Baxter equation]]:

* [[Chen Ning Yang]], M. L. Ge (eds.), *Braid Group, Knot Theory and Statistical Mechanics*, Advanced Series in Mathematical Physics **9**, World Scientific (1991) [[doi:10.1142/0796](https://doi.org/10.1142/0796)]

On the [[Aharonov-Bohm effect]], generalized to non-abelian [[Yang-Mills theory]] and its description by [[connection on a bundle|connections]] on [[fiber bundles in physics]]:

* {#WuYang75} [[Tai Tsun Wu]], [[Chen Ning Yang]], *Concept of nonintegrable phase factors and global formulation of gauge fields*, Phys. Rev. D **12** (1975) 3845 [[doi:10.1103/PhysRevD.12.3845](https://doi.org/10.1103/PhysRevD.12.3845)]

On [[Dirac monopoles]]:

* [[Tai Tsun Wu]], [[Chen Ning Yang]], _Dirac monopole without strings: monopole harmonics_, Nuclear Physics B107:3 (1976) 365--380

* [[Tai Tsun Wu]], [[Chen Ning Yang]], _Dirac's monopole without strings: Classical Lagrangian theory_, Phys. Rev. D 14, 437 (1976)

On the historical origin of [[Maxwell's equations]], the "[[vector potential]]" and ([[Yang-Mills theory|Yang-Mills]]) [[gauge theory]]:

* A. C. T. Wu, [[Chen Ning Yang]], *Evolution of the concept of vector potential in the description of the fundamental interactions*,  International Journal of Modern Physics A **21** 16 (2006) 3235-3277 [[doi:10.1142/S0217751X06033143](https://doi.org/10.1142/S0217751X06033143)]

* [[Chen Ning Yang]], *The conceptual origins of Maxwell’s equations and gauge theory*, Phyics Today **67** 11 (2014) [[doi:10.1063/PT.3.2585](https://doi.org/10.1063/PT.3.2585), [pdf](http://home.ustc.edu.cn/~lxsphys/2021-3-18/The%20conceptual%20origins%20of%20Maxwell%27s%20equations%20and%20gauge%20theory.pdf)]

## Quotes
 {#Quotes}

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On [[Yang-Mills theory]] ([[gauge theory]]) being about [[connection on a bundle|connections]] on [[principal bundles]]/[[fiber bundles]]:

Yang wrote in  _C. N. Yang, Selected papers, 1945-1980, with commentary_, W. H. Freeman and Company, San Francisco, 1983, on p. 567:

> In 1975, impressed with the fact that [[gauge fields]] are [[connection on a bundle|connections]] on [[fiber bundles]], I drove to the house of [[Shiing-shen Chern|S. S. Chern]] in El Cerrito, near Berkeley... I said I found it amazing that [[gauge theory]] are exactly [[connection on a bundle|connections]] on [[fiber bundles]], which the mathematicians developed without reference to the physical world. I added: "this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere." He immediately protested: "No, no. These concepts were not dreamed up. They were natural and real."

Yang expanded on this passage in an interview recorded as

* D. Z. Zhang,  _C. N. Yang and contemporary mathematics_, The Mathematical Intelligencer 15, 13–21 (1993) ([doi:10.1007/BF03024319](https://doi.org/10.1007/BF03024319)):

> $[$This$]$ was not just joy. There was something more, something deeper: After all, what could be more mysterious, what could be more awe-inspiring, than to find that the structure of the [[observable universe|physical world]] is intimately tied to the deep [[mathematics|mathematical]] concepts, concepts which were developed out of considerations rooted only in [[logic]] and the beauty of form?

## Related entries

* [[Yang-Mills theory]]

* [[Yang monopole]]

category: people


Chen space

{:myundef: .un_defn style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

[[Kuo Tsai Chen]] described several categories of [[generalized smooth space|generalised smooth spaces]] in his works (see also at _[[diffeological space]]_).
Four are reproduced below.

His initial motivation appears to be extending the theory of [[differential forms]] from finite-dimensional [[manifolds]] to [[path spaces]].

+-- {: myundef}
###### 1973

By a convex $n$-region (or, simply a _convex_ region), we mean a closed convex region in $\mathbb{R}^n$.  A convex $0$-region consists of a single point.

_Definition._ A differentiable space $X$ is a [[Hausdorff space]] equipped with a family of maps called plots which satisfy the following conditions:

1. Every plot is a continuous map of the type $\phi: U \to X$, where $U$ is a convex region.

2. If $U'$ is also a convex region (not necessarily of the same dimension as $U$) and if $\theta: U' \to U$ is a $C^\infty$-map, then $\phi\theta$ is also a plot.

3. Each map $\{0\} \to X$ is a plot.
=--

+-- {: myundef}
###### 1975

By a convex region we mean a closed convex set in $\mathbb{R}^n$ for some finite $n$.

__Definition.__  A predifferentiable space $X$ is a topological space equipped with a family of maps called plots which satisfy the following conditions:

1. Every plot is a continuous map of the type $\phi: U \to X$, where $U$ is a convex region.

2. If $U'$ is also a convex region (not necessarily of the same dimension as $U$) and if $\theta: U' \to U$ is a $C^\infty$-map, then $\phi\theta$ is also a plot.

3. Each map $\{0\} \to X$ is a plot.

__Remark.__ in [1973], a predifferentiable space is called a "differentiable space".
We propose to amend the definition of a differentiable space by adding the following condition:

4. Let $\phi: U\to X$ be a continuous map and let $\{\theta_i: U_i \to U\}$ be a family of $C^\infty$-maps, $U$, $U_i$ being convex regions, such that a function $f$ on $U$ is $C^\infty$ if and only if each $f\circ \theta_i$ is $C^\infty$ on $U_i$.
If each $\phi \circ \theta_i$ is a plot of $X$, then $\phi$ itself is a plot of $X$.
=--

+-- {: myundef}
###### 1977

  The symbols $U$, $U'$, $U_i$, ... will denote convex sets.
  All convex sets will be finite dimensional.
  They will serve as models, i.e. sets whose differentiable structure is known.

...

__Definition 1.2.1__ A differentiable space $M$ is a set equipped with a family of set maps called plots, which satisfy the following conditions:

1. Every plot is a map of the type $U\to M$, where $\dim U$ can be arbitrary.

2. If $\phi: U \to M$ is a plot and if $\theta: U' \to U$ is a $C^\infty$-map, then $\phi \circ \theta$ is a plot.

3. Every constant map from a convex set to $M$ is a plot.

4. Let $\phi: U\to M$ be a set map.
  If $\{U_i\}$ is an open covering of $U$ and if each restriction $\phi | U_i$ is a plot, then $\phi$ is itself a plot.
=--

##  Remarks

* In 1986, Chen gave a definition equivalent to the last.
* It seems clear from the context that in the 1975 paper Chen was first recalling the definition from the 1973 paper.  However, his recollection was not completely accurate as the underlying object was now a [[topological space]] rather than a [[Hausdorff space]].
* The forcing condition on the maps in the 1975 paper is actually stronger than that in the 1977 paper.
* The final structure is of [[sheaf|sheaves]] on a [[site]].  This is the definition used in, for example, Baez and Hoffnung [0807.1704].

## Related notions

* [[diffeological space]]

## References

* [Iterated integrals of differential forms and loop space homology](http://www.ams.org/mathscinet-getitem?mr=380859), 1973
* [Iterated integrals, fundamental groups and covering spaces](http://www.ams.org/mathscinet-getitem?mr=0377960), 1975
* [Iterated path integrals](http://www.ams.org/mathscinet-getitem?mr=0454968), 1975
* [On differentiable spaces](http://www.ams.org/mathscinet-getitem?mr=842915), 1986
* [Convenient Categories of Smooth Spaces](http://arxiv.org/abs/0807.1704v1), 0807.1704

[[!redirects Chen spaces]]


Chen-Ruan cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Chen-Ruan cohomology_ ([Chen-Ruan 00](#ChenRuan00)) of an [[orbifold]] $\mathcal{X}$ is the traditional [[orbifold cohomology]] with [[real numbers|real]] or [[complex numbers|complex]] [[coefficients]] of its [[inertia orbifold]] $I \mathcal{X}$, hence is the [[rational cohomology]]/[[real cohomology]]/[[complex cohomology]] of the underlying [[topological quotient space]] of $I \mathcal{X}$.

Chen-Ruan cohomology is motivated from the role that [[orbifolds]] play as [[target spaces]] in [[perturbative string theory]], as in the algebraic operation of orbifolds [[2d CFTs]].

## Related concepts

* [[orbifold K-theory]]

## Properties

### Relation to delocalized and Bredon equivariant cohomology

[[!include incarnations of rational equivariant topological K-theory -- table]]

\linebreak

## References

### General

The concept originates in:

* {#ChenRuan00} [[Weimin Chen]], [[Yongbin Ruan]], _A New Cohomology Theory for Orbifold_, Commun. Math. Phys. 248 (2004) 1-31 ([arXiv:math/0004129](https://arxiv.org/abs/math/0004129))

Review:

* {#ALR07} [[Alejandro Adem]], [[Johann Leida]], [[Yongbin Ruan]], _Orbifolds and Stringy Topology_, Cambridge Tracts in Mathematics **171** (2007) ([doi:10.1017/CBO9780511543081](https://doi.org/10.1017/CBO9780511543081), [pdf](http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf))

On the Chen-Ruan cohomology of [[ADE-singularities]]:

* [[Fabio Perroni]], *Orbifold Cohomology of ADE-singularities* ([arXiv:0510528](https://arxiv.org/abs/math/0510528))

* [[Fabio Perroni]], *Chen-Ruan cohomology of ADE-singularities*, International Journal of Mathematics, Vol. 18, No. 9 (2007) 1009-1059 ([arXiv:math/0605207](https://arxiv.org/abs/math/0605207), [doi:10.1142/S0129167X07004436](https://doi.org/10.1142/S0129167X07004436))

[[!include traditional orbifold cohomology -- references]]

[[!redirects Chen-Ruan orbifold cohomology]]


Chenchang Zhu

Hi, this is Chenchang Zhu or 朱晨畅 ![](https://lh5.googleusercontent.com/Upb44JaQgs7JvC1BvXma6X4-TGvxcfHM3Dfq2vcVgUo=w266-h199-p-no)

I've got tenured at [Göttingen](http://www.uni-math.gwdg.de/zhu) (since 2013), and were long time in its [Courant Center](http://www.crcg.de/wiki/User:Zhu) (from 2008), before that I was Maitre de conferences (assistant professor) in [Grenoble](http://www-fourier.ujf-grenoble.fr/~zhu/) (from 2006), post-doc in [ETH Zurich](http://www.math.ethz.ch/) (2004-2006), Ph.D. from U.C. Berkeley with advisor [Alan Weinstein](http://math.berkeley.edu/~alanw/) (2004), undergraduate study in [Bei Jing University](http://www.math.pku.edu.cn/htdocs/main.php).

***
[CV](https://www.dropbox.com/s/1edpig6rc4sy9d3/cvzhu.pdf?dl=0), [Publication/arxiv](http://arxiv.org/a/zhu_c_1), [Publication/MathSciNet](http://www.ams.org/msnmain?fn=130&form=fullsearch&preferred_language=en&Submit=Start+Search&pg4=ICN&s4=Zhu%2C+Chenchang&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&s7=&dr=all&yrop=eq&arg3=&yearRangFirst=&yearRangeSecond=&pg3=ET&s3=All&l=20&reference_lists=show&simple_headlines=full&redirect=Providence%2C+RI+USA)

***
**Courses/Seminars** (upper levels):

[[Summer 2018 math problem solving]]

[[Winter 2017-2018 seminar on higher structures (Higher Lie Groupoids)]]

[[Winter 2016-2017 seminar on higher structures]]

Summer 2016: Symplectic Geometry, using [Ana](https://people.math.ethz.ch/~acannas/)'s [book](https://people.math.ethz.ch/~acannas/Papers/lsg.pdf)

Winter 2015-2016 [[lectures on higher bundle theory]].

Spring 2015 [[seminar on Grothendieck-Teichmuller theory]].

Fall 2014 [[seminar on factorization algebras]].

***
**Academic Activities**

June 11-22, 2018, [SMS 2018, Derived and Higher Structures in Geometry and Physics](https://ncatlab.org/higherstructureconferences/show/sms2018) at Fields Institute, Toronto

May 1-May 10, Simon Center.

April 29-May 1, [Applied Category Theory](http://www.appliedcategorytheory.org/), Leiden, see old friend, John :)

July 2017, [Lie Theory and Mathematical Physics](https://ncatlab.org/higherstructureconferences/show/Lie+Theory+and+Mathematical+Physics) at MIT

16.04.2017--21.04.2017, [Banff](http://www.birs.ca/events/2017/5-day-
workshops/17w5023)

12.04.2017--15.04.2017, Toronto

17.07.2016--23.07.2016, visit [Anton](http://www.unige.ch/math/en/people/alekseev/) in Geneva

02.07.2016--09.07.2016, [Poisson 2016, ETH Zurich](http://www.unige.ch/math/Poisson2016/), if you want to [see](http://www.video.ethz.ch/conferences/2016/poisson/day2/d2eb6d7a-7dff-4fa8-ade7-2390b9299340.html?autoplay=true) how nervous I was while giving the talk :-P

22.06.2016--26.06.2016, visit [Giovanni](http://www2.math.ethz.ch/u/felder) in ETH Zurich.

10.03.2016--13.03.2016, Leuven, visit Ph.D. brother [Marco](https://perswww.kuleuven.be/~u0096206/)

07.03.2016--10.03.2016, MPI Bonn, (a super active place with infinite many mathematicians, and we made progress again in joint project with [Christian](http://people.mpim-bonn.mpg.de/blohmann/) :)

28.02.2016--05.03.2016, visit [Anton](http://www.unige.ch/math/en/people/alekseev/)'s group in Geneva (a wonderful group full of cool young people!! :)

04.01.2016--08.01.2016, [[Poisson Geometry and Mathematical Physics]] in [Chern Institute](http://www.nim.nankai.edu.cn/nim_e/) Tianjin, China

[Higher Structures in Geometry and Physics, MPI Bonn, Oct. 12--17, 2015](https://www.mpim-bonn.mpg.de/HS2015)

3 April -- 4 April, a [[mini-workshop on Poisson geometry]] in [Huazhong University of Science and Technology](http://english.hust.edu.cn/) Wuhan, China

28 Feb--5 March, going to Berkeley :).

16 February--19 February, Hamburg [Christoph Wockel's conference](http://www.math.uni-hamburg.de/ids2015/)

Urs Schreiber is visiting 29 Jan and gives a talk at [GRK colloquium] (http://www.uni-goettingen.de/de/207450.html).

[Higher Structures in Geometry and Physics, Geneva, Oct. 27--31, 2014](http://ncatlab.org/higherstructureconferences/show/higherstructure2014)
, which is one of the [series.](http://ncatlab.org/higherstructureconferences)

***
**Teaching**

* AGLA I (2017-2018 winter) (linear algebra)

***
**Academic and Nonacademic Service**

Together with Dima Dudko, we organised a course followed by participation in IMC (international math competition), and this year [our team](http://www.uni-goettingen.de/de/3240.html?cid=5571) got no. 6 worldwise!

I'm one of the people taking care of a [chinese culture association here in Gottingen](https://www.facebook.com/Chinesischer-Kulturverein-G%C3%B6ttingen-eV-1731003327220934/?skip_nax_wizard=true.) with its [Chinese version](http://blog.sina.com.cn/u/6017846563).

I'm also [Gleichstellungsbeauftragte](http://www.math.uni-goettingen.de/personen/gleichstellungsbeauftragte.html) for math institute :).

I organised [Girls Day 2016](https://www.uni-goettingen.de/en/507545.html) here in Gottingen for higher school girls.
category: people

[[!redirects Chenchang Zhu]]
[[!redirects zcc]]


Cheng space

# Cheng spaces
* table of contents
{: toc}

## Idea

A Cheng space is a version of a [[measurable space]] developed by [[Henry Cheng]] for [[constructive mathematics]].  Even in [[classical mathematics]], however, Cheng spaces are more general than standard measure spaces.  On the other hand, if we equip a measurable space with a $\sigma$-[[sigma-ideal|ideal]] of [[null subsets]] (or a $\delta$-filter of [[full subsets]]), there is no essential difference classically.

Some of the [[abstract nonsense]] below is original research (by [[Toby Bartels]] and [[Todd Trimble]]), but based heavily on Cheng\'s work as described in [[Handbook of Constructive Analysis|Bishop & Bridges]].

## Motivation

From a [[constructive mathematics|constructive]] perspective, there are a couple of related problems with the classical theory of measurable spaces.  One is that the notion of $\sigma$-[[sigma-algebra|algebra]] is highly suspicious, because it relies on an operation, [[complement]]ation, that behaves very differently in [[intuitionistic logic]].  The other is that, even you accept the definition of $\sigma$-algebra (after all, the Lebesgue-measurable sets on the real line do still form one), there may be very few measurable functions.  (It is possible in constructive mathematics that every function from the [[real line]] to itself, regardless of measurability, is [[continuous function|continuous]].)

Indeed, if we set aside the general theory of measurable spaces and simply do Lebesgue measure ad hoc in a constructive (even predicative) way, we find that instead of measurable [[functions]] we really want measurable [[partial functions]] whose domain of definition is a [[full subset]], the [[almost functions]].  This suggests that if we want to define the concept of measurable function, then we have to know what the full sets are, so we need a measurable space equipped with such a notion.  But since this is usually defined relative to a measure, well *after* the structure of the measurable space has been given, we are at an impasse.

There is a way out, due to [[Henry Cheng]], for both of these problems at once.  Instead of dealing with individual subsets, we will deal with pairs of [[disjoint subsets]].  The intuition is that we use disjoint pairs $(A,B)$ such that $A \cup B$ is full ---with $(A,\neg{A})$ being the motivating example in the classical theory---, but we let the Cheng measurability structure itself tell us which pairs those are.  Once we fix a particular measure, we may find additional pairs whose union is full, somewhat like finding additional measurable sets when taking the completion in the classical theory (although taking the completion is a separate phenomenon here), but that\'s all right; the important thing is that each pair chosen really is full in any measure used (much as each set in a classical $\sigma$-algebra must actually be measurable by any measure used).

## Definitions

Let $X$ be a [[set]].  We'll define the structure of a Cheng space on $X$ in several steps.

### The Boolean semi-algebra of disjoint pairs

A __[[disjoint subsets|disjoint pair]]__ in $X$ is a pair $(A,B)$ of [[subsets]] of $X$ such that the [[intersection]] $A \cap B$ is [[empty subset|empty]].  Every set $A$ defines a disjoint pair $[A] = (A,\neg{A})$, but many disjoint pairs are not of this form; the extreme counterexample (unless $X$ is empty) is $(\empty,\empty)$.  We order disjoint pairs by the usual order on the first component and the opposite on the second:
$$ (A,B) \subseteq (C,D) \;\Leftrightarrow\; A \subseteq C \;\wedge\; B \supseteq D .$$
Similarly, we make the usual operations on sets into operations on disjoint pairs by applying formal [[de Morgan duality]] to the second component:

*  $ \empty = [\empty] = (\empty, X) $;
*  $ X = [X] = (X, \empty) $;
*  $ (A,B) \cup (C,D) = (A \cup C, B \cap D) $;
*  $ (A,B) \cap (C,D) = (A \cap C, B \cup D) $;
*  $ \bigcup_i (A_i,B_i) = (\bigcup_i A_i, \bigcap_i B_i) $;
*  $ \bigcap_i (A_i,B_i) = (\bigcap_i A_i, \bigcup_i B_i) $.

(Note that we do not write $[A]$ as $A$ except when $A$ is given as $\empty$ or $X$, because for example, $[A \cap B] = [A] \cap [B]$, while classically valid, may fail constructively.)

These operations form the disjoint pairs into a [[lattice]]; in fact, it is both a [[complete lattice]] and a [[distributive lattice]], but it is not constructively completely distributive in either direction.  (Compare the fact that a [[power set]] is, constructively, completely distributive only in one direction, making it a [[frame]]; here the directions are mixed by the formal duality and so neither works.  On the other hand, that the power set is a frame is used to show that the infinitary operations do define disjoint pairs.)

Finally, we define the [[complement]] of $(A,B)$, not using the complements of $A$ and $B$ (which usually are not even disjoint) but instead simply by reversing them:
$$ \neg(A,B) = (B,A) .$$
Then an actual [[de Morgan duality]] holds for these operations:

*  $ \neg\bigcup_i (A_i,B_i) = \bigcap_i \neg(A_i,B_i) $;
*  $ \neg\bigcap_i (A_i,B_i) = \bigcup_i \neg(A_i,B_i) $;
*  $ \neg\neg(A,B) = (A,B) $, the famous [[double negation]] law.

We can go on to define the relative complement $(A,B) \setminus (C,D)$ and symmetric difference $(A,B) \uplus (C,D)$ in terms of complements, intersections, and unions as usual, and they obey many of the usual classical laws.  (For instance, $\uplus$ is ---through a fairly lengthy calculation--- associative, which is not constructively true of [[symmetric difference]] on a power set.)

At this point, the reader could be forgiven for thinking that we have cleverly pulled a [[Boolean algebra]] out of a mere [[Heyting algebra]], but this is not true; aside from the give-away that this lattice is not constructively completely distributive, it is not even classically a Boolean algebra.  This is because $(A,B) \cup \neg(A,B) = (A \cup B, \empty)$ (and similarly for intersection) and there is no requirement that $A \cup B = X$.  What we have instead is a complete Boolean [[rig]], aka semi-ring with unit; to keep consistent with the usual terminology of measure theory, I\'ll call such a thing a __Boolean semi-algebra__.

This is all a special case of the [[Chu construction]]; the poset of disjoint pairs in $X$ is $Chu_{TV}(P X, \empty)$, where $TV$ is the [[enriched category|enriching]] category of [[truth values]].

### The $\sigma$-semi-algebra of complemented pairs

Given a set $X$, a __$\sigma$-semi-algebra__ on $X$ is a collection $\mathcal{M}$ of disjoint pairs in $X$, called __complemented pairs__, such that:
1.  $[\empty] = (\empty,X)$ is a complemented pair;
2.  If $(A,B)$ is a complemented pair, then so is its complement $(B,A)$;
3.  If $(A_1,B_1), (A_2,B_2), (A_3,B_3), \ldots$ are complemented pairs, then so is their union $(\bigcup_i A_i, \bigcap_i B_i)$.

The arguments above that $\mathcal{M}$ is closed under countable intersections, relative complements, and symmetric differences goes through.  (We can also define analogous notions of semi-algebra, $\delta$-semi-ring, and other variations of $\sigma$-[[sigma-algebra|algebra]].)

Finally, a __Cheng measurable space__ or simply a __Cheng space__ is a set equipped with a $\sigma$-semi-algebra.

(Incidentally, the reason why we do not use the term 'measurable pair' is that $A$ and $B$ may easily both be measurable in some sense yet without having $(A,B)$ as a complemented pair.  In particular, $(\empty,\empty)$ is rarely a complemented pair ---although that is not forbidden either---, yet it is hard to call it non-measurable.)

### Measurable sets and functions

A [[subset]] $A$ of a Cheng space $X$ is __[[measurable subset|measurable]]__ if there is some complemented pair $(A,B)$.  A subset $S$ is __[[full subset|full]]__ if, for some complemented pair $(A,B)$, $S$ contains the [[union]] $A \cup B$.  Conversely, $S$ is __[[null subset|null]]__ if, for some such $(A,B)$, $S$ is disjoint from $A \cup B$.

Given two Cheng measurable spaces $X$ and $Y$, an __[[almost function]]__ from $X$ to $Y$ is a [[partial function]] from $X$ to $Y$ such that the [[domain]] of $f$ is full.  An almost function is __[[measurable function|measurable]]__ if, given any complemented pair $(C,D)$ in $Y$, there is a complemented pair $(A,B)$ such that $\{p\colon X \;|\; p \in A \iff f(p) \in C\}$ and $\{p\colon X \;|\; p \in B \iff f(p) \in D)$ are both full.

Two (measurable) functions are __[[almost equal]]__ if their [[equaliser]] is full.

Cheng spaces, measurable almost functions between them, and almost equality between them form a [[category]] $Cheng Sp$.

## Completion

A Cheng space is __[[complete measure space|complete]]__ if, whenever $(A,B)$ is a complemented pair and $A \Leftrightarrow C$ and $B \Leftrightarrow D$ are full, then $(C,D)$ is a complemented pair.  In particular, every full set and every null set is measurable.

Given any Cheng space $(X,\mathcal{M})$, its __completion__ $(X,\bar{\mathcal{M}})$ has the same [[underlying set]] $X$ but a disjoint pair $(C,D)$ is $\bar{\mathcal{M}}$-complemented iff, for some $\mathcal{M}$-complemented pair $(A,B)$, both $A \Leftrightarrow C$ and $B \Leftrightarrow D$ are $\mathcal{M}$-full.

A Cheng space is complete iff it is its own completion; the completion of any Cheng space is complete.  The [[identity function]] on $X$ is measurable in either direction between $(X,\mathcal{M})$ and $(X,\bar{\mathcal{M}})$, so they are [[isomorphic]] in $Cheng Sp$.  Accordingly, the [[full subcategory]] of $Cheng Sp$ on the complete spaces is [[equivalence of categories|equivalent]] to $Cheng Sp$ itself (via its [[inclusion functor]]).

When restricted to complete spaces, the definition of measurable function is simpler: any partial function under which the [[preimage]] of any complemented pair is complemented.

## Measures on Cheng spaces

We will define the notion of a [[positive measure]] on a Cheng space; it's straightforward to generalise to other sorts of measures as described at [[measure]].

Given a Cheng space $(X,\mathcal{M})$, a __positive measure__ on $(X,\mathcal{M})$ is a [[function]] $\mu$ from $\mathcal{M}$ to $[0,\infty]$ such that:

*  [[absolutely continuous measure|absolute continuity]]: $\mu(A,B) = 0$ if $B$ is full;
*  [[inclusion-exclusion]]: $\mu(A \cap C, B \cup D) + \mu(A \cup C, B \cap D) = \mu(A,B) + \mu(C,D)$;
*  [[subadditive function|subadditivity]]: $\mu(\bigcup_i A_i, \bigcap_i B_i) \leq \sum_i \mu(A_i,B_i)$ for an [[infinite sequence]] of complemented pairs.

We think of $\mu(A,B)$ as the measure of $A$; thanks to absolute continuity, either $A$ or $B$ alone is enough to determine $\mu(A,B)$.

[[!redirects Cheng space]]
[[!redirects Cheng spaces]]
[[!redirects Cheng measurable space]]
[[!redirects Cheng measurable spaces]]

[[!redirects boolean semialgebra]]
[[!redirects boolean semialgebras]]
[[!redirects boolean semi-algebra]]
[[!redirects boolean semi-algebras]]
[[!redirects Boolean semialgebra]]
[[!redirects Boolean semialgebras]]
[[!redirects Boolean semi-algebra]]
[[!redirects Boolean semi-algebras]]


Chenjie Wang

* [GoogleScholar page](https://scholar.google.com/citations?user=JP3DlY4AAAAJ&hl=en)

## Selected writings

On [[loop braid group]]-[[representations]] in [[model (in theoretical physics)|models]] of [[solid state physics]]:

* [[Chenjie Wang]], [[Michael Levin]], *Braiding Statistics of Loop Excitations in Three Dimensions*, Phys. Rev. Lett. **113** (2014) 080403 ([doi:10.1103/PhysRevLett.113.080403](https://doi.org/10.1103/PhysRevLett.113.080403))

category: people


Chermaine Sia

* [webpage](http://www.math.harvard.edu/~sia/)

## related $n$Lab entries

* [[topological automorphic form]]

category: people


Chern character

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Traditionally, in the strict sense of the term, the _Chern character_ is a [[universal characteristic class]] of [[vector bundles]] or equivalently of their [[topological K-theory]] classes, which is a [[rational number|rational]] combination of all [[Chern classes]].

This is a special case of the following more general construction ([[Quadratic Functions in Geometry, Topology,and M-Theory|Hopkins-Singer 02, section 4.8]]):

for $E$ a [[spectrum]] [[Brown representability theorem|representing]] a [[generalized (Eilenberg-Steenrod) cohomology theory]] there is a canonical [[localization of spectra|localization]] map

$$
  ch_E \;\colon\; E \longrightarrow E \wedge H\mathbb{R}
$$

to the [[smash product]] with the [[Eilenberg-MacLane spectrum]] over the [[real numbers]]. This represents the $E$-Chern character (see also [Bunke-Gepner 13, around def. 2.1](#BunkeGepner13)).

In the case that $E = $ [[KU]] this reproduces the traditional Chern character. (In which case this is a map from a [[complex oriented cohomology theory]] of [[chromatic level]] 1 to chromatic level 0. More generally one can also consider [[higher chromatic Chern characters]] that take values not in [[ordinary cohomology]] but in some cohomology theory of higher [[chromatic level]]. See at _[[higher chromatic Chern character]]_ for more on this.)

The Chern character $ch_E$ may be used to define [[differential cohomology]] refinements $\hat E$ of the [[cohomology theory]] $E$ by choosing a [[differential form]]-model for $E \wedge H\mathbb{R}$ ([Hopkins-Singer 02](#HopkinsSinger02), see also at _[[differential function complex]]_). In that case $ch_E$ is the [[real cohomology]] class associated to a _chern character differential form_ $CH_E$ via the [[de Rham theorem]]. Here $CH_E$ has the interpretation of being the [[curvature forms]] of the [[differential cohomology]] [[cocycles]] thought of as [[connection on a principal ∞-bundle|∞-connections]].

This may be turned around ([Bunke-Nikolaus-Völkl 13, prop. .3.5](#BunkeNikolausVoelkl13)): given any refinement $\hat E$ of $E$ in a [[tangent cohesive (∞,1)-topos]] $T \mathbf{H}$, then it is induced from homotopy pullback of its [de Rham coefficients](cohesive+%28infinity%2C1%29-topos+--+structures#deRhamCohomology) along a Chern character map

$$
  ch_E = \Pi \theta_{\hat E} \;\colon\; E \simeq \Pi(\hat E) \longrightarrow \Pi \flat_{dR} \hat E
  \,,
$$

where $\Pi$ is the [[shape modality]] and $\theta_E$ the [[Maurer-Cartan form]] of $E$. This reproduces the above definition for ordinary differential form models, see at _[differential cohomology diagram -- Hopkins-Singer coefficients](#differential+cohomology+diagram#HopkinsSingerCoefficients)_.

But more generally, given for instance a [[K(n)-local spectrum|K(n)-localization]] $E \longrightarrow L_{K(n)} E$ then any choice of [[cohesion|cohesive]] refinement of $L_{K(n)} E$ (i.e. lift through the [[unit of a monad|unit]] of the [[shape modality]] $\Pi$) which is in the kernel of $\flat$ yields a generalized differential cohomology theory $\hat E$ whose intrinsic Chern-character $\Pi \theta_{\hat E}$ is the $K(n)$-localization. See at _[differential cohomology diagram -- Chern character and differential fracture](differential+cohomology+diagram#ChernCharacterAndFractureSquares)_.

In words this is summarized succintly as: _The Chern character is the [[shape modality|shape]] of the [[Maurer-Cartan form]]._

In the context of [[algebraic K-theory]] Chern characters appear at _[[Beilinson regulators]]_.
There are analogues in algebraic geometry (e.g. a Chern character between [[Chow groups]] and [[algebraic K-theory]]) and in [[noncommutative geometry]] (Chern-Connes character) where the role of usual cohomology is taken by some variant of cyclic cohomology.

## Examples

### For vector bundles and topological K-theory
 {#KTheory}

The classical theory of the Chern character applies to the [[spectrum]] of complex [[K-theory]], $E = KU$. In this case, the Chern character is made up from Chern classes: each characteristic class is by Chern-Weil theory in  the image of a certain element in the Weil algebra via taking the class of evaluation at the [[curvature]] operator for some choice of a connection. Consider the symmetric functions in $n$ variables $t_1,\ldots, t_n$ and let the Chern classes of a complex vector bundle $\xi$ (representing a complex K-theory class) be $c_1,\ldots, c_n$. Define the formal power series

$$
\phi = \phi^n(t_1,\ldots, t_n) = e^{t_1}+\ldots+e^{t_n}= \sum_{k=0}^\infty \frac{1}{k!} (t_1^k+\ldots+t_n^k)
$$

Then $ch(\chi) = \phi(c_1,\ldots,c_n)$.

Let us describe this a bit differently. The cocycle $H^0(X,KU)$ may be represented by a complex [[vector bundle]], and the image of this cocycle under the Chern-character is the class in even-graded real cohomology that is represented (under the [[deRham theorem]] isomorphism of deRham cohomology with real cohomology) by the even graded closed [[differential form]]

$$
 ch(\nabla) \coloneqq \sum_{j \in \mathbb{N}}
   k_j
   tr( F_\nabla \wedge \cdots \wedge F_\nabla)
  \;\;
  \in \Omega^{2 \bullet}(X)
  \,,
$$

where

* $\nabla$ is any chosen [[connection on a bundle|connection]] on the vector bundle;

* $F = F_\nabla \in \Omega^2(X,End(V))$ is the [[curvature]] of this connection;

* $k_j \in \mathbb{R}$ are normalization constants, $k_j = \frac{1}{j!} \left( \frac{1}{2\pi i}\right)^j$;

* the trace of the wedge products produces the [[curvature characteristic form]]s.

The Chern character applied to the [[Whitney sum]] of two vector bundles is a sum of the Chern characters for the two: $ch(\xi\oplus \eta) = ch(\chi)+ch(\eta)$ and it is multiplicative under the tensor product of vector bundles: $ch(\xi\otimes\eta)=ch(\chi)ch(\eta)$. Therefore we get a ring homomorphism.

### For spectra and generalized cohomology theories
 {#ForSpectraAndGeneralizedCohomology}

For $E$ a [[spectrum]] and $E^\bullet$ the [[generalized (Eilenberg-Steenrod) cohomology theory|generalized cohomology theory]] it [[Brown representability theorem|represents]]

$$
  E^\bullet(X) \;\simeq\; \pi_{-\bullet} Maps(X,E)
$$

the _[[Chern-Dold character]] for $E$_ ([Buchstaber 70](#Buchstaber70)) is the map induced by [[rationalization]] over the [[real numbers]]

$$
  E \overset{L_{\mathbb{R}}}{\longrightarrow} E_{\mathbb{R}}
$$

i.e. is

\[
  \label{ChernDoldCharacter}
  chd
  \;\colon\;
  E^\bullet(X)
  \;\simeq\;
  \pi_{-\bullet}Maps(X,E)
  \overset{
    \pi_{-\bullet}Maps(X,L_{\mathbb{R}})
  }{\longrightarrow}
  \pi_{-\bullet}Maps(X,E_{\mathbb{R}})
  \;\simeq\;
  E^\bullet_{\mathbb{E}}(X)
  \;\simeq\;
  H^\bullet(X, \pi_{\bullet}(E)\otimes_{\mathbb{Z}}\mathbb{R})
  \,.
\]

The very last equivalence in (eq:ChernDoldCharacter) is due to [Dold 56, Cor. 4](#Dold56) (reviewed in detail in [Rudyak 98, II.3.17](#Rudyak98), see also  [Gross 19, Def. 2.5](#Gross19)).

One place where this neat state of affairs (eq:ChernDoldCharacter) is made fully explicit is [Lind-Sati-Westerland 16, Def. 2.1](#LindSatiWesterland16). Many other references leave this statement somewhat in between the lines (e.g. [Buchstaber 70](#Buchstaber70), [Upmeier 14](#Upmeier14)) and, in addition, often without reference to Dold (e.g. [Hopkins-Singer 02, Sec. 4.8](#HopkinsSinger02), [Bunke 12, Def. 4.45](#Bunke12), [Bunke-Gepner 13, Def. 2.1](#BunkeGepner13), [Bunke-Nikolaus 14, p. 17](#BunkeNikolaus14)).

Beware that some authors say _[[Chern-Dold character]]_ for the full map in (eq:ChernDoldCharacter) (e.g.
[Buchstaber 70](#Buchstaber70),
[Upmeier 14](#Upmeier14), [Lind-Sati-Westerland 16, Def. 2.1](#LindSatiWesterland16)), while other authors mean by this only that last equivalence in (eq:ChernDoldCharacter) (e.g. [Rudyak 98, II.3.17](#Rudyak98), [Gross 19, Def. 2.5](#Gross19)).

Examples of [[Chern-Dold characters]]:

* [[Chern character]] on [[KU]];

* [[Pontrjagin character]] on [[KO]].

### For cohesive stable homotopy types

More generally, for $\hat E$ a [[stable homotopy type]] in a [[cohesive (∞,1)-topos]], then the underlying bare homotopy type is $E \coloneqq \Pi(\hat E)$ and the corresponding Chern character is

$$
  ch \coloneqq \Pi \theta_{\hat E} \;\colon\; E \simeq \Pi(\hat E) \longrightarrow \Pi \flat_{dR} \hat E
  \,.
$$

For more on this see at _[[differential cohomology diagram]]_.

### In terms of cyclic homology

Generalizing in another direction, generalized Chern characters are given by passage to [[derived loop spaces]] and their [[cyclic homology]] or, more generally,  [[topological cyclic homology]] ([Toen-Vezzosi 08](#ToenVezzosi08), [Hoyois-Scherotzke-Sibilla 15](#HoyoisScherotzkeSibilla15)).

## Properties

### Intertwining Thom classes with Todd classes

+-- {: .num_prop #RationalToddClassIsChernCharacterOfThomClass}
###### Proposition
**([[rational Todd class is Chern character of Thom class]])**

Let $V \to X$ be a [[complex vector bundle]] over a [[compact topological space]]. Then the [[Todd class]] $Td(V) \,\in\, H^{ev}(X; \mathbb{Q})$ of $V$ in [[rational cohomology]] equals the [[Chern character]] $ch$ of the [[Thom class]] $th(V) \,\in\, K\big( Th(V) \big)$ in the [[complex topological K-theory]] of the [[Thom space]] $Th(V)$, when both are compared via the [[Thom isomorphisms]] $\phi_E \;\colon\; E(X) \overset{\simeq}{\to} E\big( Th(V)\big)$:

$$
  \phi_{H\mathbb{Q}}
  \big(
    Td(V)
  \big)
  \;=\;
  ch\big( th(V) \big)
  \,.
$$

More generally , for $x \in K(X)$ any class, we have

$$
  \phi_{H\mathbb{Q}}
  \big(
    ch(x)
    \cup
    Td(V)
  \big)
  \;=\;
  ch\big( \phi_{K}(x) \big)
  \,,
$$

which specializes to the previous statement for $x = 1$.

=--

([Karoubi 78, Chapter V, Theorem 4.4](rational+Todd+class+is+Chern+character+of+Thom+class#Karoubi78))

### Push-forward and Grothendieck-Riemann-Roch theorem

The behaviour of the Chern-character under [[fiber integration in generalized cohomology]] along [[proper maps]] is described by the [[Grothendieck-Riemann-Roch theorem]].

### Compatibility with the Adams operations
 {#CompatibilityWithTheAdamsOperations}

The [[Adams operations]] $\psi^k$ on [[complex topological K-theory]] are compatible with the [[Chern character map]] to [[rational cohomology]] in that the effect of $\psi^k$ on the Chern character image in degree $2r$ is multiplication by $k^r$:

+-- {: .num_defn #AdamsLikeOperationsOnRationalCohomology}
###### Definition
**(Adams-like operations on [[rational cohomology]])**

For $X$ a [[topological space]], with [[rational cohomology]] in even degrees denoted

$$
  H^{ev}(X;\, \mathbb{Q})
  \;\colon\;
  \underset{r \in \mathbb{N}}{\prod}
  H^{2 r}(X;\, \mathbb{Q})
$$

define graded [[linear maps]]

$$
  \psi^k_{H} \;\colon\; H^{ev}(X) \longrightarrow H^{ev}(X)
$$

for $k \in \mathbb{N}$ by taking their restriction to degree $2r$ to act by multiplication with $k^r$:

$$
  \array{
    H^{2r}(X;\mathbb{Q})
    &\overset{\;\;\;\psi^k_H\;\;\;}{\longrightarrow}&
    H^{2r}(X;\mathbb{Q})
    \\
    \alpha_{2k} &\mapsto& k^{r} \cdot \alpha
    \,.
  }
$$

=--

+-- {: .num_prop #AdamsOperationsAreCompatibleWithTheChernCharacter }
###### Proposition
**([[Adams operations compatible with the Chern character]])**

For $X$ a [[topological space]] with a finite [[CW-complex]]-[[mathematical structure]], the [[Chern character]] $ch$ on the [[complex topological K-theory]] of $X$ intertwines the [[Adams operations]] $\psi^n$ on K-theory with the Adams-like operations $\psi^n_H$ on [[rational cohomology]] from Def. \ref{AdamsLikeOperationsOnRationalCohomology}, for $k \geq 1$, in that the following [[commuting diagram|diagram commutes]]:

$$
  \array{
    K(X)
      &\overset{\;\;\;ch\;\;\;}{\longrightarrow}&
    H^{ev}(X;\,\mathbb{Q})
    \\
    {}^{ \mathllap{ \psi^k } }
    \big\downarrow
    &&
    \big\downarrow
    {}^{ \mathrlap{ \psi^k_H } }
    \\
    K(X)
      &\underset{\;\;\;ch\;\;\;}{\longrightarrow}&
    H^{ev}(X;\,\mathbb{Q})
    \,,
  }
$$

=--

([Adams 62, Thm. 5.1. (vi)](Adams+operations#Adams62), review in [Karoubi 78, Chapter V, Theorem 3.27](Adams+operations#Karoubi78), [Maakestad 06, Thm. 4.9](#Maakestad06))

+-- {: .proof}
###### Proof idea

Use the exponentional-formula for the [[Chern character]] with the  [[splitting principle]].

=--

## Related concepts

* [[odd Chern character]]

* [[twisted Chern character]]

* [[equivariant Chern character]]

* [[twisted equivariant Chern character]]

* [[Chern-Dold character]]

* [[Bismut-Chern character]]

* for [[algebraic K-theory]]:

  * [[cyclotomic trace]]

  * [[regulator]], [[Beilinson regulator]]

* [[Grothendieck-Riemann-Roch theorem]]

* [[higher chromatic Chern character]]

  * [[elliptic Chern character]]

  * [[Morava E-theoretic Chern character]]

* [[transchromatic character]]

* [[group character]]

## References

### Chern character on K-theory

Original Discussion of the Chern character on complex [[topological K-theory]]:

* {#Hirzebruch56} [[Friedrich Hirzebruch]], Section 12.1 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))

* {#BorelHirzebruch58} [[Armand Borel]], [[Friedrich Hirzebruch]], Section 9.1 in: _Characteristic Classes and Homogeneous Spaces, I_, American Journal of Mathematics Vol. 80, No. 2 (Apr., 1958), pp. 458-538 ([jstor:2372795](https://www.jstor.org/stable/2372795))

* {#AtiyahHirzebruch61} [[M. F. Atiyah]], [[F. Hirzebruch]], Section 1.10 in: _Vector bundles and homogeneous spaces_, 1961, Proc. Sympos. Pure Math., Vol. III pp. 7–38 American Mathematical Society, Providence, R.I. ([web](http://hirzebruch.mpim-bonn.mpg.de/87/), doi:10.1142/9789814401319_0008, [MR 0139181](http://www.ams.org/mathscinet-getitem?mr=0139181))

* {#Hilton71} [[Peter Hilton]], _General cohomology theory and K-theory_, London Mathematical Society Lecture Note Series 1, Cambridge University Press (1971) ([doi:10.1017/CBO9780511662577](https://doi.org/10.1017/CBO9780511662577))

* {#Karoubi78} [[Max Karoubi]], Chapter V.3 of: _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 ([pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007%2F978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3))

Further discussion

* {#Maakestad06} Helge Maakestad, _Notes on the Chern-character_, Journal of Generalized Lie Theory and Applications, 2017, 11:1 ([arXiv:math/0612060](https://arxiv.org/abs/math/0612060), [doi:10.4172/1736-4337.100025](https://www.hilarispublisher.com/open-access/notes-on-the-cherncharacter-.pdf))

* [[Goncalo Tabuada]], _A universal characterization of the Chern character maps_ ([arXiv/1002.3276](http://arxiv.org/abs/1002.3726))

Discussion of the [[equivariant Chern character]] in [[equivariant K-theory]]:

* {#Stefanich} German Stefanich, _Chern Character in Twisted and Equivariant K-Theory_ ([pdf](https://math.berkeley.edu/~germans/Chern2.pdf))

### Chern-Dold character on generalized cohomology

The identification of rational [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology]] as [[ordinary cohomology]] with [[coefficients]] in the rationalized [[stable homotopy groups]] is due to

* {#Dold56} [[Albrecht Dold]], _Relations between ordinary and extraordinary homology_, Matematika, 9:2 (1965), 8–14; Colloq. algebr. Topology, Aarhus Universitet, 1962, 2–9 ([mathnet:mat350](http://mi.mathnet.ru/eng/mat350)), reprinted in: J. Adams & G. Shepherd (Authors), _Algebraic Topology: A Student's Guide_ (London Mathematical Society Lecture Note Series, pp. 166-177). Cambridge: Cambridge University Press 1972 ([doi:10.1017/CBO9780511662584.015](https://doi.org/10.1017/CBO9780511662584.015))

reviewed in

* [Hilton 71, Theorem 3.18](#Hilton71)

* {#Rudyak98} [[Yuli Rudyak]], II.7.13 in: _On Thom Spectra, Orientability, and Cobordism_, Springer 1998 ([doi:10.1007/978-3-540-77751-9](https://doi.org/10.1007/978-3-540-77751-9))

* {#Gross19} Jacob Gross, _The homology of moduli stacks of complexes_ ([arXiv:1907.03269](https://arxiv.org/abs/1907.03269))

The combination of [Dold 56](#Dold56) to the [[Chern-Dold character]] on [[generalized (Eilenberg-Steenrod) cohomology theory]] is due (for [[complex cobordism cohomology]]) to

* {#Buchstaber70} [[Victor Buchstaber]], _The Chern–Dold character in cobordisms. I_,

  Russian original: Mat. Sb. (N.S.), 1970 Volume 83(125), Number 4(12), Pages 575–595 ([mathnet:3530](http://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=3530&option_lang=eng))

  English translation: Mathematics of the USSR-Sbornik, Volume 12, Number 4, AMS 1970 ([doi:10.1070/SM1970v012n04ABEH000939](https://iopscience.iop.org/article/10.1070/SM1970v012n04ABEH000939))

Review in

* [Hilton 71, p. 50](#Hilton71)

That the Chern-Dold character reduces to the original Chern character on K-theory is

* [Hilton 71, Theorem 5.8](#Hilton71).

That the Chern-Dold character is given by rationalization of representing spectra is made fully explicit in

* {#LindSatiWesterland16} [[John Lind]], [[Hisham Sati]], [[Craig Westerland]], Section 2.1: _Twisted iterated algebraic K-theory and topological T-duality for sphere bundles_, Ann. K-Th. 5 (2020) 1-42 ([arXiv:1601.06285](https://arxiv.org/abs/1601.06285))

This rationalization construction appears also (without attribution to [#Hilton 71](#Hilton71) or [Buchstaber 70](#Buchstaber70) or [Dold 56](#Dold56)) in the following articles (all in the context of [[differential cohomology]]):

* {#HopkinsSinger02} [[Mike Hopkins]], [[Isadore Singer]], Section [4.8, page 47](http://arxiv.org/PS_cache/math/pdf/0211/0211216v2.pdf#page=47) of _[[Quadratic Functions in Geometry, Topology,and M-Theory]]_, ([math.AT/0211216](http://arxiv.org/abs/math.AT/0211216)).

* {#Bunke12} [[Ulrich Bunke]], _Differential cohomology_ ([arXiv:1208.3961](http://arxiv.org/abs/1208.3961))

* {#BunkeGepner13} [[Ulrich Bunke]], [[David Gepner]], around def. 2.1 of: _Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory_ ([arXiv:1306.0247](http://arxiv.org/abs/1306.0247))

* {#Upmeier14} Markus Upmeier, _Refinements of the Chern-Dold Character: Cocycle Additions in Differential Cohomology_,  J. Homotopy Relat. Struct. 11, 291–307 (2016). ([arXiv:1404.2027](https://arxiv.org/abs/1404.2027), [doi:10.1007/s40062-015-0106-y](https://doi.org/10.1007/s40062-015-0106-y))

* {#BunkeNikolaus14} [[Ulrich Bunke]], [[Thomas Nikolaus]], _Twisted differential cohomology_, Algebr. Geom. Topol. Volume 19, Number 4 (2019), 1631-1710. ([arXiv:1406.3231](http://arxiv.org/abs/1406.3231), [euclid:euclid.agt/1566439272](https://projecteuclid.org/euclid.agt/1566439272))

More on the [[Chern-Dold character]] on [[complex cobordism cohomology]]:

* [[Victor Buchstaber]], A. P. Veselov, _Chern-Dold character in complex cobordisms and abelian varieties_ ([arXiv:2007.05782](https://arxiv.org/abs/2007.05782))

The observation putting this into the general context of [[differential cohomology diagrams]] (see there) of [[stable homotopy types]] in [[cohesion]] is due to

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], section 4.4. of _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

based on [Bunke-Gepner 13](#BunkeGepner13).

Further generalization of the [[Chern-Dold character]] to [[non-abelian cohomology]]:

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The Character Map in Nonabelian Cohomology --- Twisted, Differential, Generalizaed]]_, World Scientific (2023) [[arXiv:2009.11909](https://arxiv.org/abs/2009.11909), [doi:10.1142/13422](https://doi.org/10.1142/13422)]

The [[equivariant Chern-Dold character]] in [[equivariant cohomology]]:

* [[Wolfgang Lück]], _Chern characters for proper equivariant homology theories and applications to K- and L-theory_, Journal für die reine und angewandte Mathematik, Volume 2002: Issue 543 ([doi:10.1515/crll.2002.015](https://doi.org/10.1515/crll.2002.015), [pdf](https://www.him.uni-bonn.de/lueck/data/eh.pdf))

* [[Wolfgang Lück]], _Equivariant Cohomological Chern Characters_, International Journal of Algebra and Computation, Vol. 15, No. 05n06, pp. 1025-1052 (2005) ([arXiv:math/0401047](https://arxiv.org/abs/math/0401047), [doi:10.1142/S0218196705002773](https://doi.org/10.1142/S0218196705002773))

* [[Wolfgang Lück]], _Equivariant Chern characters_, 2006 ([pdf](https://www.him.uni-bonn.de/lueck/data/goettingen_equi_Chern061130trans.pdf), [[LuckEquivariantChernCharacters.pdf:file]])

### In derived algebraic geometry

Discussion of Chern characters in terms of [[free loop space objects]] in [[derived geometry]]:

* {#ToenVezzosi08} [[Bertrand Toën]], [[Gabriele Vezzosi]], _A note on Chern character, loop spaces and derived algebraic geometry_ ([arXiv:0804.1274](http://arxiv.org/abs/0804.1274))

which conjectures a construction that is fully developed in

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Caractères de Chern, traces équivariantes et géométrie algébrique dérivée_ ([arXiv:0903.3292](http://arxiv.org/abs/0903.3292))

See also

* [[Marc Hoyois]], _Chern character and derived algebraic geometry_ (2009) ([pdf](http://math.mit.edu/~hoyois/papers/chern.pdf))

* {#HoyoisScherotzkeSibilla15} [[Marc Hoyois]], [[Sarah Scherotzke]], [[Nicolò Sibilla]], _Higher traces, noncommutative motives, and the categorified Chern character_ ([arXiv:1511.03589](http://arxiv.org/abs/1511.03589))

[[!redirects Chern characters]]


Chern class

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The ordinary _Chern classes_ are the [[integral cohomology|integral]] [[characteristic classes]]

$$
  c_i : B U \to B^{2 i} \mathbb{Z}
$$

of the [[classifying space]] $B U$ of the [[unitary group]].

Accordingly these are characteristic classes in [[ordinary cohomology]] of [[unitary group|U]]-[[principal bundles]] and hence of [[complex vector bundle]]

The first Chern class is the unique characteristic class of [[circle group]]-principal bundles.

The analogous classes for the [[orthogonal group]] are the [[Pontryagin classes]].

More generally, there are _[[generalized Chern classes]]_ for any [[complex oriented cohomology theory]] ([Adams 74](#Adams74), [Lurie 10](#Lurie10)).

## Definition

+-- {: .num_defn}
###### Definition

For $n \geq 1$ the _[[universal characteristic classes|universal]] Chern [[characteristic classes|classes]]_

$$
  c_i
    \;\in\;
  H^{2i}
  \big(
    B U(n), \mathbb{Z}
  \big)
$$

of the [[classifying space]] $B U(n)$ of the [[unitary group]] are the [[cohomology classes]] of $B U(n)$ in [[integral cohomology]] that are characterized as follows:

1. $c_0 = 1$ and $c_i = 0$ if $i \gt n$;

1. for $n = 1$, $c_1$ is the canonical generator of $H^2(B U(1), \mathbb{Z})\simeq \mathbb{Z}$;

1. under pullback along the inclusion $i : B U(n) \to B U(n+1)$ we have $i^* c_i^{(n+1)} = c_i^{(n)}$;

1. under the inclusion $B U(k) \times B U(l) \to B U(k+l)$ we have $i^* c_i = \sum_{j = 0}^i c_i \cup c_{j-i}$.

The corresponding _total Chern class_ is the formal sum

$$
  c
  \;\coloneqq\;
  1 + c_1 + c_2 + \cdots
  \;\in\;
  \underset{k}{\prod}
  H^{2k}
  \big(
    B U(n)
  \big)
$$

=--

## Properties

### Existence

+-- {: .num_prop #GeneratorsOfCohomologyOfBunChernClasses}
###### Proposition

The [[cohomology ring]] of the [[classifying space]] $B U(n)$ (for the [[unitary group]] $U(n)$) is the [[polynomial ring]] on generators $\{c_k\}_{k = 1}^{n}$ of degree 2, called the _Chern classes_

$$
  H^\bullet(B U(n), \mathbb{Z})
    \simeq
  \mathbb{Z}[c_1, \cdots, c_n]
  \,.
$$

Moreover, for $B i \colon B U(n_1) \longrightarrow BU(n_2)$ the canonical inclusion for $n_1 \leq n_2 \in \mathbb{N}$, then the induced pullback map on cohomology

$$
   (B i)^\ast
     \;\colon\;
   H^\bullet(B U(n_2))
     \longrightarrow
   H^\bullet(B U(n_1))
$$

is given by

$$
  (B i)^\ast(c_k)
  \;=\;
  \left\{
    \array{
      c_k & for \; 1 \leq k \leq n_1
      \\
      0 & otherwise
    }
  \right.
  \,.
$$

=--

(e.g. [Kochman 96, theorem 2.3.1](#Kochman96))

+-- {: .proof}
###### Proof

For $n = 1$, in which case $B U(1) \simeq \mathbb{C}P^\infty$ is the infinite [[complex projective space]], we have ([prop](complex+projective+space#OrdinaryCohomologyOfComplexProjectiveSpace))

$$
  H^\bullet(B U(1)) \simeq \mathbb{Z}[ c_1 ]
  \,,
$$

where $c_1$ is the [[first Chern class]]. From here we proceed by [[induction]]. So assume that the statement has been shown for $n-1$.

Observe that the canonical map $B U(n-1) \to B U(n)$ has as [[homotopy fiber]]  the [[n-sphere|(2n-1)sphere]] ([prop.](classifying+space#SphereFibrationOverInclusionOfClassifyingSpaces)) hence there is a [[homotopy fiber sequence]] of the form

$$
  S^{2n-1} \longrightarrow B U(n-1) \longrightarrow B U(n)
  \,.
$$

Consider the induced [[Thom-Gysin sequence]].

In odd degrees $2k+1 \lt 2n$ it gives the [[exact sequence]]

$$
  \cdots
   \to
  H^{2k}(B U(n-1))
   \longrightarrow
  \underset{\simeq 0}{\underbrace{H^{2k+1-2n}(B U(n))}}
   \longrightarrow
  H^{2k+1}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  \underset{\simeq 0}{\underbrace{H^{2k+1}(B U(n-1))}}
  \to
  \cdots
  \,,
$$

where the right term vanishes by induction assumption, and the middle term since [[ordinary cohomology]] vanishes in negative degrees. Hence

$$
  H^{2k+1}(B U(n)) \simeq 0 \;\;\; for \; 2k+1 \lt 2n
$$

Then for $2k+1 \gt 2n$ the Thom-Gysin sequence gives

$$
  \cdots
   \to
  H^{2k+1-2n}(B U(n))
   \longrightarrow
  H^{2k+1}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  \underset{\simeq 0}{\underbrace{H^{2k+1}(B U(n-1))}}
  \to
  \cdots
  \,,
$$

where again the right term vanishes by the induction assumption. Hence [[exact sequence|exactness]] now gives that

$$
  H^{2k+1-2n}(B U(n))
   \overset{}{\longrightarrow}
  H^{2k+1}(B U(n))
$$

is an [[epimorphism]], and so with the previous statement it follows that

$$
  H^{2k+1}(B U(n)) \simeq 0
$$

for all $k$.

Next consider the Thom Gysin sequence in degrees $2k$

$$
  \cdots
   \to
  \underset{\simeq 0}{\underbrace{H^{2k-1}(B U(n-1))}}
   \longrightarrow
  H^{2k-2n}(B U(n))
   \longrightarrow
  H^{2k}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  H^{2k}(B U(n-1))
   \longrightarrow
  \underset{\simeq 0}{\underbrace{H^{2k +1 - 2n}(B U(n))}}
  \to
  \cdots
  \,.
$$

Here the left term vanishes by the induction assumption, while the right term vanishes by the previous statement. Hence we have a [[short exact sequence]]

$$
  0
   \to
  H^{2k-2n}(B U(n))
   \longrightarrow
  H^{2k}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  H^{2k}(B U(n-1))
    \to
  0
$$

for all $k$. In degrees $\bullet\leq 2n$ this says

$$
  0
   \to
  \mathbb{Z}
   \overset{c_n \cup (-)}{\longrightarrow}
  H^{\bullet \leq 2n}(B U(n))
   \overset{(B i)^\ast}{\longrightarrow}
  (\mathbb{Z}[c_1, \cdots, c_{n-1}])_{\bullet \leq 2n}
    \to
  0
$$

for some [[Thom class]] $c_n \in H^{2n}(B U(n))$, which we identify with the next Chern class.

Since [[free abelian groups]] are [[projective objects]] in [[Ab]], their [[extensions]] are all split (the [[Ext]]-group out of them vanishes), hence the above gives a [[direct sum]] decomposition

$$
  \begin{aligned}
    H^{\bullet \leq 2n}(B U(n))
    & \simeq
    (\mathbb{Z}[c_1, \cdots, c_{n-1}])_{\bullet \leq 2n} \oplus \mathbb{Z}\langle 2n\rangle
    \\
    & \simeq
    (\mathbb{Z}[c_1, \cdots, c_{n}])_{\bullet \leq 2n}
  \end{aligned}
  \,.
$$

Now by another induction over these short exact sequences, the claim follows.

=--

### First Chern class

* The [[first Chern class]] of a bundle $P$ is the class of its [[determinant line bundle]] $det P$

  $$
    c_1(P) = [det P]
    \,.
  $$

  See [[determinant line bundle]] for more.

### Top Chern class
 {#TopChernClass}

For $\mathcal{V}_X$ a [[complex vector bundle]] of complex [[rank]] $n$, the highest degree [[Chern class]] that may generally be non-vanishing is $c_n$. This is hence often called the _[[top Chern class]]_ of the vector bundle.

\begin{prop}
  The [[top Chern class]] of a [[complex vector bundle]] $\mathcal{V}_X$ equals the [[Euler class]] $e$ of the underlying [[real vector bundle]] $\mathcal{V}^{\mathbb{R}}_X$:

$$
  \mathcal{V}_X
  \;
  \text{has complex rank}\;n
  \;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;
  c_n
  \big(
    \mathcal{V}_X
  \big)
  \;\;
    =
  \;\;
  e
  \big(
    \mathcal{V}^{\mathbb{R}}_X
  \big)
  \;\;\;\;
  \in
  H^{2n}
  \big(
   X;
   \,
   \mathbb{Z}
  \big)
  \,.
$$
\end{prop}

(e.g. [Bott-Tu 82 (20.10.6)](#BottTu82))

\begin{prop}
  The [[top Chern class]] of a [[complex vector bundle]] $\mathcal{V}_X$ equals the [[pullback in cohomology|pullback]] of any [[Thom class]]
$th \;\in\; H^{2n}\big( \mathcal{V}_X; \mathbb{Z} \big)$ on $\mathcal{V}_X$ along the [[zero-section]]:

$$
  \mathcal{V}_X
  \;
  \text{has complex rank}\;n
  \;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;
  c_n
  \big(
    \mathcal{V}_X
  \big)
  \;\;\;
    =
  \;\;\;
  (0_X)^\ast
  (th)
  \;\;
  \in
  \;
  H^{2n}
  \big(
    X ;
    \,
    \mathbb{Z}
  \big)
$$
\end{prop}

(e.g. [Bott-Tu 82, Prop. 12.4](#BottTu82))

### Splitting principle and Chern roots
 {#SplittingPrinciple}

Under the [[splitting principle]] all Chern classes are determined by [[first Chern classes]]:

Write $i \colon T \simeq U(1)^n \hookrightarrow U(n)$ for the [[maximal torus]] inside the [[unitary group]], which is the [[subgroup]] of [[diagonal matrix|diagonal]] unitary matrices. Then

$$
  H^\bullet(B T, \mathbb{Z}) \simeq H^\bullet(B U(1)^n, \mathbb{Z})
$$

is the [[polynomial ring]] in $n$ [[generators]] (to be thought of as the universal [[first Chern classes]] $c_i$ of each copy of $B U(1)$; equivalently as the [weights](group+character#RelationToChernRootsAndSplittingPrinciple) of the [[group characters]] of $U(n)$) which are traditionally written $x_i$:

$$
  H^\bullet(B U(1)^n, \mathbb{Z})  \simeq \mathbb{Z}[x_1, \cdots, x_n]
  \,.
$$

Write

$$
  B i \;\colon\; B U(1)^n \to B U(n)
$$

for the induced map of [[deloopings]]/[[classifying spaces]], then the $k$-universal Chern class $c_k \in H^{2k}(B U(n), \mathbb{Z})$ is uniquely characterized by the fact that its pullback to $B U(1)^n$ is the $k$th [[elementary symmetric polynomial]] $\sigma_k$ applied to these first Chern classes:

$$
 (B i)^\ast (c_k) = \sigma_k(x_1, \cdots, x_n)
  \,.
$$

Equivalently, for $c = \sum_{i = 1}^n c_k$ the formal sum of all the Chern classes, and using the fact that the [[elementary symmetric polynomials]] $\sigma_k(x_1, \cdots, k_n)$ are the degree-$k$ piece in $(1+x_1) \cdots (1+x_n)$, this means that

$$
 (B i)^\ast (c) = (1+x_1) (1+ x_2) \cdots (1+ x_n)
  \,.
$$

Since here on the right the first Chern classes $x_i$ appear as the [[roots]] of the Chern polynomial, they are also called **Chern roots**.

See also at _[splitting principle -- Examples -- Complex vector bundles and their Chern roots](splitting+principle#ComplexVectorBundleAndTheirChernRoots)_.

(e.g. [Kochman 96, theorem 2.3.2](#Kochman96), [tom Dieck 08, theorem 19.3.2](#tomDieck08))

+-- {: .num_lemma #FromBUnTOBU1nPullbackInCohomologyIsInjective}
###### Lemma

For $n \in \mathbb{N}$ let $B \iota_n \;\colon\; B (U(1)^n) \longrightarrow B U(n)$ be the canonical map. Then the induced pullback operation on [[ordinary cohomology]]

$$
  \left(
    B \iota_n
  \right)
  \;\colon\;
   H^\bullet( B U(n); \mathbb{Z} )
   \longrightarrow
   H^\bullet( B U(1)^n; \mathbb{Z} )
$$

is a [[monomorphism]].

=--

A **[[proof]]** of lemma \ref{FromBUnTOBU1nPullbackInCohomologyIsInjective}, via analysis of the [[Serre spectral sequence]] of $U(n)/U(1)^n \to B U(1)^n \to B U(n)$ is indicated in ([Kochman 96, p. 40](#Kochman96)). A proof via [[Becker-Gottlieb transfer|transfer]] of the [[Euler class]] of $U(n)/U(1)^n$, following ([Dupont 78, (8.28)](#Dupont78)), is indicated at _[[splitting principle]]_ ([here](splitting+principle#InjectivityOfPullbackInCohomologyToBT)).

+-- {: .num_prop #SplittingPrincipleForChernClasses}
###### Proposition

For $k \leq n \in \mathbb{N}$ let $B \iota_n \;\colon\; B (U(1)^n) \longrightarrow B U(n)$ be the canonical map. Then the induced pullback operation on [[ordinary cohomology]] is of the form

$$
  (B i_n)^\ast
  \;\colon\;
  \mathbb{Z}[c_1, \cdots, c_k]
  \longrightarrow
  \mathbb{Z}[(c_1)_1,\cdots (c_1)_n]
$$

and sends the $k$th Chern class $c_k$ (def. \ref{GeneratorsOfCohomologyOfBunChernClasses}) to the $k$th [[elementary symmetric polynomial]] in the $n$ copies of the [[first Chern class]]:

$$
  (B i_n)^\ast
    \;\colon\;
  c_k \mapsto \sigma_k( (c_1)_1, \cdots, (c_1)_n )
  \,.
$$

=--

+-- {: .proof}
###### Proof

First consider the case $n = 1$.

The [[classifying space]] $B U(1)$ is equivalently the infinite [[complex projective space]] $\mathbb{C}P^\infty$. Its [[ordinary cohomology]] is the [[polynomial ring]] on a single generator $c_1$, the [[first Chern class]] ([prop.](complex+projective+space#OrdinaryCohomologyOfComplexProjectiveSpace))

$$
  H^\bullet(B U(1))
    \simeq
  \mathbb{Z}[ c_1 ]
  \,.
$$

Moreover, $B i_1$ is the identity and the statement follows.

Now by the [[Künneth theorem]] for ordinary cohomology ([prop.](K%C3%BCnneth+theorem#KunnethInOrdinaryCohomology)) the cohomology of the [[Cartesian product]] of $n$ copies of $B U(1)$ is the [[polynomial ring]] in $n$ generators

$$
  H^\bullet(B U(1)^n)
   \simeq
  \mathbb{Z}[(c_1)_1, \cdots, (c_1)_n]
  \,.
$$

By prop. \ref{GeneratorsOfCohomologyOfBunChernClasses} the domain of $(B i_n)^\ast$ is the [[polynomial ring]] in the Chern classes $\{c_i\}$, and by the previous statement the codomain is the polynomial ring on $n$ copies of the first Chern class

$$
  (B i_n)^\ast
    \;\colon\;
   \mathbb{Z}[ c_1, \cdots, c_n ]
     \longrightarrow
   \mathbb{Z}[ (c_1)_1, \cdots, (c_1)_n ]
  \,.
$$

This allows to compute $(B i_n)^\ast(c_k)$ by [[induction]]:

Consider $n \geq 2$ and assume that $(B i_{n-1})^\ast_{n-1}(c_k) = \sigma_k((c_1)_1, \cdots, (c_1)_{(n-1)})$. We need to show that then also $(B i_n)^\ast(c_k) = \sigma_k((c_1)_1,\cdots, (c_1)_n)$.

Consider then the [[commuting diagram]]

$$
  \array{
    B U(1)^{n-1}
      &\overset{ B i_{n-1} }{\longrightarrow}&
    B U(n-1)
    \\
    {}^{\mathllap{B j_{\hat t}}}\downarrow
      &&
    \downarrow^{\mathrlap{B i_{\hat t}}}
    \\
    B U(1)^n
     &\underset{B i_n}{\longrightarrow}&
    B U(n)
  }
$$

where both vertical morphisms are induced from the inclusion

$$
  \mathbb{C}^{n-1} \hookrightarrow \mathbb{C}^n
$$

which omits the $t$th coordinate.

Since two embeddings $i_{\hat t_1}, i_{\hat t_2} \colon U(n-1) \hookrightarrow U(n)$ differ by [[conjugation]] with an element in $U(n)$, hence by an [[inner automorphism]], the maps $B i_{\hat t_1}$ and $B_{\hat i_{t_2}}$ are [[homotopy|homotopic]], and hence $(B i_{\hat t})^\ast = (B i_{\hat n})^\ast$, which is the morphism from prop. \ref{GeneratorsOfCohomologyOfBunChernClasses}.

By that proposition, $(B i_{\hat t})^\ast$ is the identity on $c_{k \lt n}$ and hence by induction assumption

$$
  \begin{aligned}
    (B i_{n-1})^\ast (B i_{\hat t})^\ast c_{k \lt n}
    &=
    (B i_{n-1})^\ast c_{k \lt n}
    \\
    =
    \sigma_k( (c_1)_1, \cdots, \widehat{(c_1)_t}, \cdots, (c_1)_n )
  \end{aligned}
  \,.
$$

Since pullback along the left vertical morphism sends $(c_1)_t$ to zero and is the identity on the other generators, this shows that

$$
  (B i_n)^\ast(c_{k \lt n})
    \simeq
   \sigma_{k\lt n}((c_1)_1, \cdots, \widehat{(c_1)_t}, \cdots, (c_1)_n)
   \;\;
   mod (c_1)_t
  \,.
$$

This implies the claim for $k \lt n$.

For the case $k = n$ the commutativity of the diagram and the fact that the right map is zero on $c_n$ by prop. \ref{GeneratorsOfCohomologyOfBunChernClasses} shows that the element $(B j_{\hat t})^\ast (B i_n)^\ast c_n = 0$ for all $1 \leq t \leq n$. But by lemma \ref{FromBUnTOBU1nPullbackInCohomologyIsInjective} the morphism $(B i_n)^\ast$, is injective, and hence $(B i_n)^\ast(c_n)$ is non-zero. Therefore for this to be annihilated by the morphisms that send $(c_1)_t$ to zero, for all $t$, the element must be proportional to all the $(c_1)_t$. By degree reasons this means that it has to be the product of all of them

$$
  \begin{aligned}
    (B i_n)^{\ast}(c_n)
     & = (c_1)_1 \otimes (c_1)_2 \otimes \cdots \otimes (c_1)_n
     \\
     & = \sigma_n( (c_1)_1, \cdots, (c_1)_n )
  \end{aligned}
  \,.
$$

This completes the induction step.

=--

### Whitney sum formula
 {#WhitneySumFormula}

+-- {: .num_prop #WhitneySumChernClasses}
###### Proposition

For $k\leq n \in \mathbb{N}$, consider the canonical map

$$
  \mu_{k,n-k}
    \;\colon\;
  B U(k) \times B U(n-k)
    \longrightarrow
  B U(n)
$$

(which classifies the [[Whitney sum]] of [[complex vector bundles]] of [[rank]] $k$ with those of rank $n-k$). Under pullback along this map the universal [[Chern classes]] (prop. \ref{GeneratorsOfCohomologyOfBunChernClasses}) are given by

$$
  (\mu_{k,n-k})^\ast(c_t)
  \;=\;
  \underoverset{i = 0}{t}{\sum} c_i \otimes c_{t-i}
  \,,
$$

where we take $c_0 = 1$ and  $c_j = 0 \in H^\bullet(B U(r))$ if $j \gt r$.

So in particular

$$
  (\mu_{k,n-k})^\ast(c_n) \;=\; c_k \otimes c_{n-k}
  \,.
$$

=--

e.g. ([Kochman 96, corollary 2.3.4](#Kochman96))

+-- {: .proof}
###### Proof

Consider the [[commuting diagram]]

$$
  \array{
    H^\bullet( B U(n) )
      &\overset{\mu_{k,n-k}^\ast}{\longrightarrow}&
    H^\bullet( B U(k) ) \otimes H^\bullet( B U(n-k) )
    \\
    {}^{\mathllap{\mu_k^\ast}}\downarrow
      &&
    \downarrow^{\mathrlap{ \mu_{k}^\ast \otimes \mu_{n-k}^\ast }}
    \\
    H^\bullet( B U(1)^n )
     &\simeq&
    H^\bullet( B U(1)^k ) \otimes H^\bullet( B U(1)^{n-k} )
  }
  \,.
$$

This says that for all $t$ then

$$
  \begin{aligned}
    (\mu_k^\ast \otimes \mu_{n-k}^\ast) \mu_{k,n-k}^\ast(c_t)
    & =
    \mu^\ast_n(c_t)
    \\
    & = \sigma_t((c_1)_1, \cdots, (c_1)_n)
  \end{aligned}
  \,,
$$

where the last equation is by prop. \ref{SplittingPrincipleForChernClasses}.

Now the [[elementary symmetric polynomial]] on the right decomposes as required by the left hand side of this equation as follows:

$$
  \sigma_t((c_1)_1, \cdots, (c_1)_n)
    \;=\;
  \underoverset{r = 0}{t}{\sum}
  \sigma_r((c_1)_1, \cdots, (c_1)_{n-k})
  \cdot
  \sigma_{t-r}( (c_1)_{n-k+1}, \cdots, (c_1)_n )
  \,,
$$

where we agree with $\sigma_q((c_1)_1, \cdots, (c_1)_p) = 0$ if $q \gt p$. It follows that

$$
  (\mu_k^\ast \otimes \mu_{n-k}^\ast) \mu_{k,n-k}^\ast(c_t)
  =
  (\mu_k^\ast \otimes \mu_{n-k}^\ast)
  \left(
    \underoverset{r=0}{t}{\sum} c_r \otimes c_{t-r}
  \right)
  \,.
$$

Since $(\mu_k^\ast \otimes \mu_{n-k}^\ast)$ is a monomorphism by lemma \ref{FromBUnTOBU1nPullbackInCohomologyIsInjective}, this implies the claim.

=--

[[!include Chern- and Pontrjagin forms -- section]]

## Examples

### Chern classes of linear representations
 {#ChernClassesOfLinearRepresentations}

Under the [[Atiyah-Segal completion]] map [[linear representations]] of a [[group]] $G$ induce K-theory classes on the [[classifying space]] $B G$. Their Chern classes are hence invariants of the [[linear representations]] themselves.

See at _[[characteristic class of a linear representation]]_ for more.

## Related concepts

* [[Chern root]]

* [[Pontryagin class]]

* [[Stiefel-Whitney class]]

* [[Chern character]]

* [[Chern insulator]]

In [[Yang-Mills theory]] field configurations with non-vanishing [[second Chern class]] (and minimal energy) are called [[instantons]]. The second Chern class is the _[[instanton number]]_ . For more on this see at _[SU(2)-instantons from the correct maths to the traditional physics story](BPST-instanton#FromTheMathsToThePhysicsStory)_.

## References

Original articles:

* {#Hirzebruch56} [[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. Grothendieck]], _La théorie des classes de Chern_, Bulletin de la Société Mathématique de France __86__ (1958), p. 137--154, ([numdam:BSMF_1958__86__137_0](http://www.numdam.org/item?id=BSMF_1958__86__137_0))

Textbook accounts:

* {#KobayashiNomizu63} [[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))

* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], chapter IX of volume II of _[[Connections, Curvature, and Cohomology]]_ Academic Press (1973)

* [[John Milnor]], [[Jim Stasheff|James D. Stasheff]], _Characteristic Classes_, Annals of Mathematics Studies 76, Princeton University Press (1974) ([ISBN:9780691081229](https://press.princeton.edu/books/paperback/9780691081229/characteristic-classes-am-76-volume-76), [doi:10.1515/9781400881826](https://doi.org/10.1515/9781400881826), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/milnstas.pdf))

* {#BottTu82} [[Raoul Bott]], [[Loring Tu]], _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Mathematics 82, Springer 1982 ([doi:10.1007/BFb0063500](https://doi.org/10.1007/BFb0063500))

* {#Kochman96} [[Stanley Kochman]], section 2.3 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#tomDieck08} [[Tammo tom Dieck]], _Algebraic topology_, EMS 2008

With an eye towards [[mathematical physics]]:

* [[Mikio Nakahara]], Section 11.3 of: *[[Geometry, Topology and Physics]]*, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), <a href="http://alpha.sinp.msu.ru/~panov/LibBooks/GRAV/(Graduate_Student_Series_in_Physics)Mikio_Nakahara-Geometry,_Topology_and_Physics,_Second_Edition_(Graduate_Student_Series_in_Physics)-Institute_of_Physics_Publishing(2003).pdf">pdf</a>)

* {#RudolphSchmidt} [[Gerd Rudolph]], [[Matthias Schmidt]], Def. 4.2.2 of _Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields_, Theoretical and Mathematical Physics series, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

A brief introduction is in chapter 23, section 7

* [[Peter May]], _A concise course in algebraic topology_ ([pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

For [[Conner-Floyd Chern classes]] in [[complex oriented cohomology theory]]:

* {#Adams74} [[Frank Adams]], part II.2 and part III.10 of _[[Stable homotopy and generalised homology]]_, 1974

* {#Lurie10} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, 2010, lecture 4 ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture4.pdf)) and lecture 5 ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture5.pdf))

See also

* {#Dupont78} Dupont, _Curvature and characteristic classes_, Springer 1978

* 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>

[[!redirects Chern classes]]

[[!redirects universal Chern class]]
[[!redirects universal Chern classes]]

[[!redirects second Chern class]]

[[!redirects Chern root]]
[[!redirects Chern roots]]

[[!redirects total Chern class]]
[[!redirects total Chern classes]]


Chern connection

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[holomorphic vector bundle]]

## References

Lecture notes:

* {#Moroianu07} [[Andrei Moroianu]], Thm. 4.3 in: _Lectures on Kähler Geometry_, Cambridge University Press 2007 [[arXiv:math/0402223](https://arxiv.org/abs/math/0402223) [doi:10.1017/CBO9780511618666](https://doi.org/10.1017/CBO9780511618666), [pdf](https://moroianu.perso.math.cnrs.fr/tex/kg.pdf)]

* {#Geal08} geal2004@cms.zju.edu.cn: *Chern connections and Chern curvatures* (2008) [[[ChernConnections.pdf:file]]]

See also:

* Wikipedia, *[Hermitian metrics on a holomorphic vector bundle](https://en.wikipedia.org/wiki/Holomorphic_vector_bundle#Hermitian_metrics_on_a_holomorphic_vector_bundle)*

* Nefton Pali, *Chern connections and Chern curvature of the tangent bundle of almost complex manifolds* [[arXiv:math/0402028](https://arxiv.org/abs/math/0402028)]

Discussion for [[holomorphic bundle gerbes]] with an eye twoards [[string 2-connections]]:

* Roberto Tellez-Dominguez, *Chern correspondence for higher principal bundles* [[arXiv:2310.12738](https://arxiv.org/abs/2310.12738)]

[[!redirects Chern connections]]


Chern insulator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### Chern insulators

In [[solid state physics]], a [[topological insulator]] which is *not* "[[symmetry protected topological phase|protected by any symmetry]]" has a [[valence bundle]] $V$ over the [[Brillouin torus]] $\mathbb{R}^d$ which is a [[complex vector bundle]] unconstrained by any reality or other conditions, and hence whose class in [[complex K-theory]] $KU^0(\mathbb{T}^d)$ is thought to classify the [[topological phase of matter|topological phase]] of the "unprotected" [[topological insulator]], according to the expected [[K-theory classification of topological phases of matter]].

Now the [[Brillouin torus]] of a realistic material is of effective [[dimension of a manifold|dimension]] $d \leq 3$ and over spaces of these dimensions the only [[characteristic class]] of a [[complex vector bundle]] is its [[first Chern class]]  $c_1(V) \,\in\, H^2\big(\mathbb{T}^d; \, \mathbb{Z}\big)$. Specifically for effectively 2-dimensional materials, the first *Chern number* $c_1[V] = \int_{\mathbb{T}^2} c_1(V) \,\in\, \mathcal{C}$ coincides with this K-theory class:

\begin{tikzcd}[row sep=2pt, column sep=0pt]
  KU^0
  \big(
    \mathbb{T}^2_\ast
  \big)
  &\simeq&
  KU^0
  \big(
    S^2_\ast
  \big)
  &\simeq&
  \mathbb{Z}
  \\
  {[V]}
  \ar[rrrr, |->]
  &&&&
  c_1[V]
\end{tikzcd}

For this reason such "symmetry un-protected" topological insulators are commonly known as *Chern insulators* and the [[first Chern class|first Chern number]] of their [[valence bundle]] is regarded as labeling their [[topological phases of matter|topological phase]].

### Haldane model & QAHE

While mathematically the case without any quantum symmetry is the simplest one, physically this typically has to be realized with more effort by starting with a symmetry-protected material (such as [[graphene]]) and then "[[symmetry breaking|breaking]]" its symmetries (either by manipulating the material in the lab or, quite often, just theoretically by adding more terms to the Hamiltonian of its theoretical model).

Therefore it is regarded as a seminal achievement when the *[[Haldane model]]* was found, which *intrinsically* breaks the symmetries of an idealized  graphene-like model (by [[spin-orbit coupling]]) and which is generally regarded as the archetype of a (model for a) Chern insulator.

An *extrinsic* way to break the [[time-reversal symmetry]] of a material like [[graphene]] is to simply place it in an external [[magnetic field]]; in this case one sometimes speaks of a *[[quantum Hall material]]*, instead.

With reference to this relation, the intrinsic effect that breaks the [[time-reversal symmetry]] of some material to a Chern insulator-phase are also known as a *[[quantum anomalous Hall effect]]* (QAHE).

### Chern semi-metals

If a Chern insulator is deformed (mentally at least) to a [[semi-metal]] with band gap closures over isolated nodal points $\{k_1, \cdots, k_N\}$ in the [[Brillouin torus]], then the relevant domain space becomes the [[complement]] $\mathbb{T}^3 \setminus \{k_1, \cdots, k_N\}$, which admits non-trivial Chern numbers for each puncture, corresponding to the evaluation of $c_1(V)$ on any [[2-sphere]] surrounding the $I$th nodal point. This now measures the topological protection of the semimetal phase (see eg. [Mathai-Thiang 17b](semi-metal#MathaiThiang17b) for a transparent account).

Some authors speak of "Chern semi-metals" to amplify this.

## Related concepts

* [[topological insulator]]

* [[K-theory classification of topological phases of matter]]

## References

### General

* {#Vanderbilt18} [[David Vanderbilt]],  Section 5.1 of: *Berry Phases in Electronic Structure Theory -- Electric Polarization, Orbital Magnetization and Topological Insulators*, Cambridge University Press (2018) ([doi:10.1017/9781316662205](https://doi.org/10.1017/9781316662205))

* Panagiotis Kotetes, Chapter 5 of: *Topological Insulators*, IOP Science 2019 ([ISBN:978-1-68174-517-6](https://iopscience.iop.org/book/978-1-68174-517-6))

* N. Regnault, B. Andrei Bernevig, *Fractional Chern Insulator*, Phys. Rev. X 1, 021014 (2011) ([arXiv:1105.4867](https://arxiv.org/abs/1105.4867))

Models akin to the [[Haldane model]]:

* Doru Sticlet, [[Frédéric Piéchon]], [[Jean-Noël Fuchs]], Pavel Kalugin, Pascal Simon, *Geometrical engineering of a two-bands Chern insulator in two dimensions with arbitrary topological index*, Phys. Rev. B **85** 165456 (2012) [[arXiv:1201.6613](https://arxiv.org/abs/1201.6613), [doi:10.1103/PhysRevB.85.165456](https://doi.org/10.1103/PhysRevB.85.165456)]

* [[Marwa Mannaï]], [[Jean-Noël Fuchs]], [[Frédéric Piéchon]], [[Sonia Haddad]], *Stacking-induced Chern insulator* [[arXiv:2208.02491](https://arxiv.org/abs/2208.02491)]

### Experimental realization

* Aizhu Wang, Xiaoming Zhang, Yuanping Feng, Mingwen Zhao*, *Chern Insulator and Chern Half-Metal States in the Two-Dimensional Spin-Gapless Semiconductor $Mn_2 \mathrm{C}_6\mathrm{S}_12$*, J. Phys. Chem. Lett. **8** 16 (2017) 3770–3775 $[$[doi:10.1021/acs.jpclett.7b01187](https://doi.org/10.1021/acs.jpclett.7b01187)$]$

[[!redirects Chern insulators]]

[[!redirects quantum anomalous Hall effect]]

[[!redirects Chern semimetal]]
[[!redirects Chern semimetals]]

[[!redirects Chern semi-metal]]
[[!redirects Chern semi-metals]]


Chern- and Pontrjagin forms -- section

### Chern-, Pontrjagin-, and Euler- characteristic forms

We spell out the formulas for the images under the [[Chern-Weil homomorphism]] of the [[Chern classes]], [[Pontrjagin classes]] and [[Euler classes]] as [[characteristic forms]] over [[smooth manifolds]].

#### Preliminaries

Let $X$ be a [[smooth manifold]].

Write

\[
  \label{RingOfEvenDegreeDifferentialForms}
  \Omega^{2\bullet}(X)
  \;\;
  \in
  \;
  CAlg_{\mathbb{R}}
\]

for the [[commutative algebra]] over the [[real numbers]] of [[even number|even]]-degree [[differential forms]] on $X$, under the [[wedge product]] of differential forms. This is naturally a [[graded commutative algebra]], graded by form degree, but since we consider only forms in even degree it is actually a plain [[commutative algebra]], too, after forgetting the grading.

Let $\mathfrak{g}$ be a [[semisimple Lie algebra]] (such as [[special unitary Lie algebra|$\mathfrak{su}(d)$]] or [[special orthogonal Lie algebra|$\mathfrak{so}(d)$]]) with [[Lie algebra representation]] $V \,\in\, Rep_{\mathbb{C}}(\mathfrak{g})$ over the [[complex numbers]] of [[finite number|finite]] [[dimension of a vector space|dimension]] $dim_{\mathbb{C}}(V) \,=\, n \,\in\, \mathbb{N}$ (for instance the [[adjoint representation]] or the [[fundamental representation]]), hence a [[homomorphism]] of [[Lie algebras]]

$$
  \mathfrak{g}
  \xrightarrow{\;\;\rho\;\;}
  End_{\mathbb{C}}(V)
$$

to the [[linear map|linear]] [[endomorphism ring]] $End_{\mathbb{C}}(V)$, regarded here through its [[commutator]] as the [[endomorphism Lie algebra]] of $V$.

When regarded as an [[associative ring]] this is [[isomorphism|isomorphic]] to the [[matrix algebra]] of $n \times n$ [[square matrices]]

\[
  \label{LinearEndomorphismRing}
  End_{\mathbb{C}}(V)
  \;\;
  \simeq
  \;\;
  Mat_{n \times n}(\mathbb{C})
  \,.
\]

The [[tensor product of vector spaces|tensor product]] of the $\mathbb{C}$-[[associative algebra|algebras]] (eq:RingOfEvenDegreeDifferentialForms) and (eq:LinearEndomorphismRing)

is equivalently the $n \times n$ [[matrix algebra]] with [[coefficients]] in the [[complexification]] of even-degree differential forms:
$$
  \Omega^{2\bullet}
  \big(X\big)
  \otimes_{\mathbb{R}}
  End_{\mathbb{C}}(V)
  \;\simeq\;
  \Omega^{2\bullet}(X)
  \otimes_{\mathbb{R}}
  \big(
    Mat_{n \times n}( \mathbb{R} )
  \big)
  \;\;
  \simeq
  \;\;
  Mat_{n \times n}
  \big(
    \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} \mathbb{C}
  \big)
  \,.
$$

The multiplicative [[unit]]

\[
  \label{UnitInAlgebraOfEndomorphismValuedEvenDegreeDifferentialForms}
  I
  \;\in\;
  Mat_{n \times n}
  \big(
    \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} \mathbb{C}
  \big)
\]

in this algebra is the [[smooth function]] ([[differential 0-forms]]) which is [[constant function|constant]] on the $n \times n$ [[identity matrix]] and independent of $t$.

Given a [[connection on a principal bundle|connection]] on a $G$-[[principal bundle]], we regard its [[Lie algebra valued differential form|$\mathfrak{g}$-valued]] [[curvature form]] as an element of this algebra

\[
  \label{CurvatureFormAsElementOfMatrixValuedEvenDegreeDifferentialForm}
  F_\nabla
    \,\in\,
  \Omega^2(X) \otimes_{\mathbb{R}} \mathfrak{g}
  \xrightarrow{\; \rho \;}
  \Omega^2(X) \otimes_{\mathbb{R}} End_{\mathbb{C}}(V)
  \xhookrightarrow{\;\;\;}
  \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} End_{\mathbb{C}}(V)[t]
  \;\simeq\;
  Mat_{n \times n}
  \Big(
    \mathbb{C} \otimes_{\mathbb{R}} \Omega^{2}(X)
  \Big)
  \,.
\]

#### The formulas

##### Chern forms

The *[[total Chern form]]* $c(\nabla)$ is the [[determinant]] of the [[sum]] of the unit (eq:UnitInAlgebraOfEndomorphismValuedEvenDegreeDifferentialForms) with the curvature form (eq:CurvatureFormAsElementOfMatrixValuedEvenDegreeDifferentialForm),
and its component in degree $2k$, for $k \in \mathbb{N}$, is the *$k$th [[Chern form]]* $c_k(\nabla)$:

$$
  c(\nabla)
  \;\;
    \coloneqq
  \;\;
  \sum_k
  \underset{
    \mathclap{
      deg = 2k
    }
  }{
  \underbrace{
    c_k(\nabla)
  }
  }
  \;\;
    \coloneqq
  \;\;
  det
  \left(
    I + t \frac{i F_\nabla}{2\pi}
  \right)
  \,.
$$

By the [[relation between determinant and trace]], this is equal to the [[exponential]] of the [[trace]] of the [[logarithm]] of $I + \frac{i F_\nabla}{2\pi}$, this being the [[exponential series]] in the [[trace]] of the [[Mercator series]] in $\frac{i F_\nabla}{2\pi}$:

\[
  \label{ExplicitUnwindingOfTotalChernForm}
  \begin{aligned}
  c(\nabla)
  &
  \;=\;
  det
  \left(
    I + t \frac{i F_\nabla}{2\pi}
  \right)
  \\
  &
  \;=\;
  \exp
  \circ
    tr
    \circ
      ln
      \left(
        I + \frac{i F_\nabla}{2\pi}
      \right)
  \\
  &
  \;=\;
  \exp
  \circ
    tr
    \left(
      -
      \underset
        {k \in \mathbb{N}_+}
        {\sum}
        \tfrac{1}{k}
        \left(
          \frac{F_\nabla}{2\pi i}
        \right)^k
    \right)
    \\
    &
    \;=\;
    \exp
    \left(
      \underset
        {k \in \mathbb{N}_+}
        {\sum}
        \tfrac{1}{k}
        \left(
          \frac
            { - (-i)^k }
            {(2\pi)^k}
          tr\big( F_\nabla^{\wedge_k} \big)
        \right)
    \right)
    \\
    &
    \;=\;
    1
    \\
    & \phantom{\;=\;}
    +
    \phantom{\frac{1}{1}}
    \left(
      i
      \tfrac{ tr\big(F_\nabla\big) }{2 \pi}
      +
      \tfrac{1}{2}
      \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2}
      -i
      \tfrac{1}{3}
      \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3}
      -
      \tfrac{1}{4}
      \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4}
      +
      \cdots
    \right)
    \\
    & \phantom{\;=\;}
    +
    \frac{1}{2}
    \left(
      i
      \tfrac{ tr\big(F_\nabla\big) }{2 \pi}
      +
      \tfrac{1}{2}
      \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2}
      -i
      \tfrac{1}{3}
      \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3}
      -
      \tfrac{1}{4}
      \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4}
      +
      \cdots
    \right)^2
    \\
    & \phantom{\;=\;}
    +
    \frac{1}{6}
    \left(
      i
      \tfrac{ tr\big(F_\nabla\big) }{2 \pi}
      +
      \tfrac{1}{2}
      \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2}
      -i
      \tfrac{1}{3}
      \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3}
      -
      \tfrac{1}{4}
      \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4}
      +
      \cdots
    \right)^3
    \\
    & \phantom{\;=\;}
    +
    \frac{1}{24}
    \left(
      i
      \tfrac{ tr\big(F_\nabla\big) }{2 \pi}
      +
      \tfrac{1}{2}
      \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2}
      -i
      \tfrac{1}{3}
      \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3}
      -
      \tfrac{1}{4}
      \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4}
      +
      \cdots
    \right)^4
    \\
    & \phantom{\;=\;}
    +
    \cdots
    \\
    & \;=\;
    1
    \\
    & \phantom{\;=\;}
    +
    i
    \frac
      { tr\big(F_\nabla\big) }
      { 2 \pi }
    \\
    & \phantom{\;=\;}
    +
    \tfrac{1}{2}
    \frac
      { tr\big( (F_\nabla)^2 \big) }
      { (2 \pi)^2 }
    +
    \frac{1}{2}
    \left(
      i
      \frac
        { tr\big( F_\nabla \big) }
        { 2\pi }
    \right)^2
    \\
    & \phantom{\;=\;}
    -
    i
    \tfrac{1}{3}
    \frac
      { tr\big( (F_\nabla)^3 \big) }
      { (2 \pi)^3 }
    +
    \frac{1}{2}
    \left(
      2
      \left(
        i
        \frac
          { tr\big( F_\nabla \big) }
          { 2 \pi }
      \right)
      \left(
        \tfrac{1}{2}
        \frac
          { tr\big( (F_\nabla)^2 \big) }
          { (2 \pi)^2 }
      \right)
    \right)
    +
    \frac{1}{6}
    \left(
      \left(
        i
        \frac
          { tr\big(F_\nabla\big) }
          { 2\pi }
      \right)^3
    \right)
    \\
    & \phantom{\;=\;}
    -
    \tfrac{1}{4}
    \frac
      {tr\big( (F_\nabla)^4 \big)}
      { (2 \pi)^4 }
    +
    \frac{1}{2}
    \left(
      \tfrac{1}{2}
      \frac
        {tr\big( (F_\nabla)^2 \big)}
        { (2 \pi)^2 }
    \right)^2
    +
    \frac{1}{24}
    \left(
      i
      \frac
        {tr\big( F_\nabla \big)}
        { 2\pi }
    \right)^4
    \\
    & \phantom{\;=\;}
    + \cdots
    \\
    & \;=\;
    1
    \\
    & \phantom{\;=\;}
    +
    \underset{ \color{blue} = c_1(\nabla) }{
    \underbrace{
    i
    \frac
      { tr\big(F_\nabla\big) }
      { 2 \pi }
    }}
    \\
    & \phantom{\;=\;}
    +
    \underset{ \color{blue} = c_2(\nabla) }{
    \underbrace{
    \frac
      {\tr\big( (F_\nabla)^2 \big) - \big( tr(F_\nabla) \big)^2  }
      { 8 \pi^2  }
    }}
    \\
    & \phantom{\;=\;}
    +
    \underset{ \color{blue} = c_3(\nabla) }{
    \underbrace{
    i
    \frac
      {
        - 2
        \cdot
        tr\big( (F_\nabla)^3  \big)
        + 3
        \cdot
        tr(F_\nabla)
        \cdot
        tr\big( (F_\nabla)^2 \big)
        -
        \big( tr(F_\nabla ) \big)^3
      }
      {48 \pi^3}
    }}
    \\
    & \phantom{\;=\;}
    +
    \underset{ \color{blue} = c_4(\nabla) }{
    \underbrace{
    \frac
      {
        -6
        \cdot
        tr\big( (F_\nabla)^4 \big)
        + 3
        \cdot
        tr\big(  (F_\nabla)^2 \big)^2
        +
        \big( tr(F_\nabla) \big)^4
      }
      {384 \pi^4}
    }}
    \\
    & \phantom{\;=\;}
    + \cdots
  \end{aligned}
\]

##### Pontrjagin forms

Setting $tr(F_\nabla) = 0$ in these expressions (eq:ExplicitUnwindingOfTotalChernForm) yields the *[[total Pontrjagin form]]* $p(\nabla)$ with degree=$4k$-components the [[Pontrjagin forms]] $p_{k}(\nabla)$:

$$
  \begin{aligned}
    p(\nabla)
    &
    \;\coloneqq\;
    \underset{k \in \mathbb{N}}{\sum}
    \underset{
      deg = 4k
    }{
      \underbrace{
        (-1)^{k}
        p_{k}(\nabla)
      }
    }
    \\
    & \;=\;
    \underset{k \in \mathbb{N}}{\sum}
    \underset{
      deg = 4k
    }{
      \underbrace{
        c_{2k}(\nabla)
      }
    }
    \\
    & \;=\;
    1
    \\
    &
    \phantom{\;=\;}
    +
    \underset{ \color{blue} = - p_1(\nabla) }{
    \underbrace{
    \frac
      {\tr\big( (F_\nabla)^2 \big) }
      { 8 \pi^2  }
    }}
    \\
    &
    \phantom{\;=\;}
    +
    \underset{ \color{blue} = p_2(\nabla) }{
    \underbrace{
    \frac
      {
        - 2
        \cdot
        tr\big( (F_\nabla)^4 \big)
        +
        tr\big(  (F_\nabla)^2 \big)^2
      }
      {128 \pi^4}
    }}
    \\
    \phantom{\;=\;}
    +
    \cdots
  \end{aligned}
$$

Hence the first couple of [[Pontrjagin forms]] are

$$
  \begin{aligned}
    p_1(\nabla)
    &
    \;=\;
    -
    \frac
      {\tr\big( (F_\nabla)^2 \big) }
      { 8 \pi^2  }
    \\
    p_2(\nabla)
    & \;=\;
    \frac
      {
        tr\big(  (F_\nabla)^2 \big)^2
        - 2
        \cdot
        tr\big( (F_\nabla)^4 \big)
      }
      {128 \pi^4}
      \,.
  \end{aligned}
$$

(See also, e.g., [Nakahara 2003, Exp. 11.5](#Nakahara03))

##### Euler forms

For $n = 2k$ and
with the [[curvature form]] again regarded as a 2-form valued $(2k) \times (2k)$-[[square matrix]]

$$
  F_{\nabla}
  \;=\;
  \big(
    (F_{\nabla})^a{}_b
  \big)_{1 \leq a,b, \leq 2k}
$$

the [[Euler form]] is its [[Pfaffian]] of this matrix, hence the following [[sum]] over [[permutations]] $\sigma \in Sym(2k)$ with summands signed by the the [[signature of a permutation|signature]] $sgn(\sigma) \in \{\pm 1\}$:

$$
  \chi_{2k}(\nabla)
  \;=\;
  \frac
    {(-1)^k}
    { (4 \pi)^k \cdot k! }
  \underset{\sigma}{\sum}
    sgn(\sigma)
    \cdot
    (F_{\nabla})_{\sigma(1)\sigma(2)}
    \wedge
    (F_{\nabla})_{\sigma(3)\sigma(4)}
    \wedge
      \cdots
    \wedge
    (F_{\nabla})_{\sigma(2k-1)\sigma(2k)}
  \,.
$$

The first of these is, using the [[Einstein summation convention]] and the [[Levi-Civita symbol]]:

$$
  \chi_4(\nabla)
  \;=\;
  \frac
    { \epsilon^{ a b c d} (F_{\nabla})_{a b} \wedge (F_\nabla)_{c d}  }
    {32 \pi^2}
$$

(See also, e.g., [Nakahara 2003, Exp. 11.7](#Nakahara03))