View source for Chern class

A [[Characteristic class|characteristic class]] defined for complex vector bundles. A Chern class of the complex vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220301.png" /> over a base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220302.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220303.png" /> and is defined for all natural indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220304.png" />. By the complete Chern class is meant the inhomogeneous characteristic class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220305.png" />, and the Chern polynomial is the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220306.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220307.png" /> is a formal unknown. Chern classes were introduced in [[#References|[1]]].

The characteristic classes, defined for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220308.png" />-dimensional complex vector bundles and with values in the integral cohomology, are naturally identified with the elements of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220309.png" />. In this sense the Chern classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203010.png" /> can be thought of as elements of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203011.png" />, the complete Chern class as an element of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203012.png" />, and the Chern polynomial as an element of the formal power series ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203013.png" />.

The Chern classes satisfy the following properties, which uniquely determine them. 1) For two vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203014.png" /> with a common base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203016.png" />, in other words <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203017.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203018.png" />. 2) For the one-dimensional universal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203019.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203020.png" /> the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203021.png" /> holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203022.png" /> is the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203023.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203024.png" /> is the [[Thom space|Thom space]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203025.png" />, which, being complex, has a uniquely-defined orientation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203026.png" />).

Consequences of the properties 1)–2) are: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203027.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203028.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203030.png" /> is the trivial bundle. The latter fact allows one to define Chern classes as elements of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203031.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203032.png" /> is a collection of non-negative integers, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203033.png" /> denotes the characteristic class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203035.png" />.

Under the natural monomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203036.png" /> induced by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203037.png" />, the Chern classes are mapped into the elementary symmetric functions, and the complete Chern class is mapped to the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203038.png" />. The image of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203039.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203040.png" /> is the subring consisting of all symmetric formal power series. Every symmetric formal power series in the Wu generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203041.png" /> determines a characteristic class that can be expressed in terms of Chern classes. For example, the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203042.png" /> determines a characteristic class with rational coefficients, called the Todd class and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203043.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203044.png" /> be a set of non-negative integers. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203045.png" /> denote the characteristic class defined by the smallest symmetric polynomial in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203047.png" />, containing the monomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203048.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203049.png" /> be an oriented multiplicative cohomology theory. Then the Chern classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203050.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203051.png" /> satisfy, as do ordinary Chern classes, the properties: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203055.png" /> is the orientation of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203056.png" />, and these properties completely determine them. As with ordinary Chern classes, one uses the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203058.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203059.png" /> are two complex vector bundles, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203060.png" /></td> </tr></table>

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where the summation is taken over all sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203062.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203063.png" />.

In place of the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203064.png" /> one may take a unitary [[Cobordism|cobordism]] theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203065.png" /> or [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203066.png" />-theory]]. For a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203067.png" />-theory the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203068.png" /> is defined by the identity mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203069.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203070.png" />-theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203071.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203072.png" /> is the Bott periodicity operator. The notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203073.png" /> is retained for Chern classes with values in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203074.png" />-theory, while Chern classes with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203075.png" />-theory are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203076.png" />.

According to the general theory, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203077.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203078.png" /> is a vector bundle with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203079.png" />. However <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203080.png" />-theory is often conveniently thought of as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203081.png" />-graded theory, identifying the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203083.png" /> via the periodicity operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203084.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203086.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203087.png" />. From this point of view it makes sense to consider, instead of the complete Chern class, the Chern polynomial

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203088.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203089.png" /> be a cohomology operation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203090.png" />-theory (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203091.png" /> terms). The polynomial

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203092.png" /></td> </tr></table>

satisfies, as does <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203093.png" />, the multiplicative property

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There is the following connection between these polynomials:

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Here both parts of the equation lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203097.png" /> is the trivial bundle of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203098.png" />. The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203099.png" /> in this construction are different from those constructed by M.F. Atiyah, who defined them by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030100.png" />. R. Stong [[#References|[2]]] defined classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030101.png" /> that satisfy the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030102.png" /></td> </tr></table>

The difference arises because, for Stong,

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The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030104.png" /> are connected with the notion of a Landweber–Novikov algebra, which is very fruitful in homotopy theory. For an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030105.png" /> of non-negative integers, consider the characteristic class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030106.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030107.png" />. There is a [[Thom isomorphism|Thom isomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030109.png" /> is the spectrum corresponding to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030110.png" />-theory. The image of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030111.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030112.png" /> determines a [[Cohomology operation|cohomology operation]] in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030113.png" />-theory. The subalgebra of the [[Steenrod algebra|Steenrod algebra]] in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030114.png" />-theory generated by the operations of this form is called the Landweber–Novikov algebra. The operation constructed from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030115.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030116.png" />.

For one-dimensional bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030117.png" /> there is the identity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030118.png" /></td> </tr></table>

This important property, which enables one to define the [[Chern character|Chern character]], does not hold in generalized cohomology theories. However there exists a formal power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030119.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030120.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030121.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030122.png" /> is the first Chern class with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030123.png" />. For the unitary cobordism theory

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030124.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030125.png" /> is the cobordism class of the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030126.png" />. This series is called the Mishchenko series.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.S. Chern,   "Characteristic classes of Hermitian manifolds"  ''Ann. of Math.'' , '''47''' :  1  (1946)  pp. 85–121</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Stong,   "Notes on cobordism theory" , Princeton Univ. Press  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R.S. Palais,   "Seminar on the Atiyah–Singer index theorem" , Princeton Univ. Press  (1965)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.E. Conner,   E.E. Floyd,   "Differentiable periodic maps" , Springer  (1964)</TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top">  M.F. Atiyah,   I.M. Singer,   "The index of elliptic operators I"  ''Ann. of Math. (2)'' , '''87'''  (1968)  pp. 484–530</TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top">  M.F. Atiyah,   G.B. Segal,   "The index of elliptic operators II"  ''Ann. of Math. (2)'' , '''87'''  (1968)  pp. 531–545</TD></TR><TR><TD valign="top">[5c]</TD> <TD valign="top">  M.F. Atiyah,   I.M. Singer,   "The index of elliptic operators III"  ''Ann. of Math. (2)'' , '''87'''  (1968)  pp. 546–604</TD></TR><TR><TD valign="top">[5d]</TD> <TD valign="top">  M.F. Atiyah,   I.M. Singer,   "The index of elliptic operators IV"  ''Ann. of Math. (2)'' , '''93'''  (1971)  pp. 119–138</TD></TR><TR><TD valign="top">[5e]</TD> <TD valign="top">  M.F. Atiyah,   I.M. Singer,   "The index of elliptic operators V"  ''Ann. of Math. (2)'' , '''93'''  (1971)  pp. 139–149</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  F. Hirzebruch,   "Topological methods in algebraic geometry" , Springer  (1978)  (Translated from German)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  D. Husemoller,   "Fibre bundles" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.M. Bukhshtaber,   "The Chern–Dold character in cobordisms"  ''Math. USSR-Sb.'' , '''12''' :  4  (1970)  pp. 573–594  ''Mat. Sb.'' , '''83'''  (1970)  pp. 575–595</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  S.P. Novikov,   "The method of algebraic topology from the viewpoint of cobordism theory"  ''Math. USSR-Izv.'' , '''4''' :  1  (1967)  pp. 827–913  ''Izv. Akad. SSSR Ser. Mat.'' , '''31''' :  4  (1967)  pp. 855–951</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M.F. Atiyah,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030127.png" />-theory: lectures" , Benjamin  (1967)</TD></TR></table>



====Comments====
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030128.png" /> denotes the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030129.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030130.png" />.

The power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030131.png" /> for a complex oriented cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030132.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030133.png" /> is the logarithm of the formal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030134.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030135.png" />; cf. [[Cobordism|Cobordism]] and [[Formal group|Formal group]] for some more details.