Namespaces
Variants
Actions

Difference between revisions of "Becker-Gottlieb transfer"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
Line 1: Line 1:
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b1102001.png" />-mapping from the base to the total space of a [[Fibration|fibration]]. It thus induces homomorphisms on homology and cohomology, which are also called Becker–Gottlieb transfers.
+
<!--
 +
b1102001.png
 +
$#A+1 = 27 n = 0
 +
$#C+1 = 27 : ~/encyclopedia/old_files/data/B110/B.1100200 Becker\ANDGottlieb transfer
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b1102002.png" /> is a Hurewicz fibration (cf. [[Covering homotopy|Covering homotopy]]) whose fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b1102003.png" /> is homotopy equivalent to a compact [[CW-complex|CW-complex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b1102004.png" />. The Becker–Gottlieb transfer is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b1102006.png" />-mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b1102007.png" />, which means that there is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b1102008.png" /> defined between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b1102009.png" />th suspensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020011.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020012.png" /> (cf. [[Suspension|Suspension]]). Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020013.png" /> induces a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020014.png" /> on any homology theory and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020015.png" /> on any cohomology theory.
+
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
An  $  S $-
 +
mapping from the base to the total space of a [[Fibration|fibration]]. It thus induces homomorphisms on homology and cohomology, which are also called Becker–Gottlieb transfers.
 +
 
 +
Suppose that $  p : E \rightarrow B $
 +
is a Hurewicz fibration (cf. [[Covering homotopy|Covering homotopy]]) whose fibre $  F $
 +
is homotopy equivalent to a compact [[CW-complex|CW-complex]] $  F $.  
 +
The Becker–Gottlieb transfer is an $  S $-
 +
mapping $  \tau : B \rightarrow E $,  
 +
which means that there is a mapping $  \tau : {\Sigma  ^ {N} B } \rightarrow {\Sigma  ^ {N} E } $
 +
defined between the $  N $
 +
th suspensions of $  B $
 +
and $  E $
 +
for some $  N $(
 +
cf. [[Suspension|Suspension]]). Thus, $  \tau $
 +
induces a homomorphism $  \tau _ {*} $
 +
on any homology theory and $  \tau  ^ {*} $
 +
on any cohomology theory.
  
 
This gives striking relations on homology and cohomology, respectively:
 
This gives striking relations on homology and cohomology, respectively:
  
<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/b/b110/b110200/b11020016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
p _ {*} \circ \tau _ {*} = \chi ( F ) ,  \tau  ^ {*} \circ p  ^ {*} = \chi ( F ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020017.png" /> denotes multiplication by the Euler–Poincaré number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020018.png" />.
+
where $  \chi ( F ) $
 +
denotes multiplication by the Euler–Poincaré number of $  F $.
  
The Becker–Gottlieb transfer was discovered in the mid-1970s. It generalizes the transfer for finite covering spaces (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020019.png" /> is a finite set of points), which had been well-known since the 1940s and which was a generalization of a group theory transfer from the 1920s.
+
The Becker–Gottlieb transfer was discovered in the mid-1970s. It generalizes the transfer for finite covering spaces ( $  F $
 +
is a finite set of points), which had been well-known since the 1940s and which was a generalization of a group theory transfer from the 1920s.
  
 
Since the discovery of the Becker–Gottlieb transfer, other transfers have been discovered which satisfy equations (a1) with the Euler–Poincaré number replaced by another elementary topological invariant. Thus,
 
Since the discovery of the Becker–Gottlieb transfer, other transfers have been discovered which satisfy equations (a1) with the Euler–Poincaré number replaced by another elementary topological invariant. Thus,
  
<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/b/b110/b110200/b11020020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
p _ {*} \circ \tau _ {*} = k, \quad \tau  ^ {*} \circ p  ^ {*} = k,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020021.png" /> denotes multiplication by an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020022.png" />, which can be a Lefschetz number, a coincidence number, a fixed-point index, a vector field index, or an intersection number. For any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020023.png" />, the greatest common divisor of all the integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020024.png" /> associated to a transfer is the Brouwer degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020025.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020027.png" /> are manifolds of the same dimension (cf. also [[Degree of a mapping|Degree of a mapping]]). Some of these transfers are induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110200/b11020028.png" />-mappings, and others are not.
+
where $  k $
 +
denotes multiplication by an integer $  k $,  
 +
which can be a Lefschetz number, a coincidence number, a fixed-point index, a vector field index, or an intersection number. For any mapping $  p : X \rightarrow Y $,  
 +
the greatest common divisor of all the integers $  k $
 +
associated to a transfer is the Brouwer degree of $  p $
 +
when $  X $
 +
and $  Y $
 +
are manifolds of the same dimension (cf. also [[Degree of a mapping|Degree of a mapping]]). Some of these transfers are induced by $  S $-
 +
mappings, and others are not.
  
 
Several mathematicians contributed to the discovery of these other transfers, most of which either generalize some or all of the Becker–Gottlieb transfer. Dold's fixed-point transfer was discovered independently of the Becker–Gottlieb transfer, which it generalizes. It is sometimes called the Becker–Gottlieb–Dold transfer.
 
Several mathematicians contributed to the discovery of these other transfers, most of which either generalize some or all of the Becker–Gottlieb transfer. Dold's fixed-point transfer was discovered independently of the Becker–Gottlieb transfer, which it generalizes. It is sometimes called the Becker–Gottlieb–Dold transfer.

Revision as of 10:33, 29 May 2020


An $ S $- mapping from the base to the total space of a fibration. It thus induces homomorphisms on homology and cohomology, which are also called Becker–Gottlieb transfers.

Suppose that $ p : E \rightarrow B $ is a Hurewicz fibration (cf. Covering homotopy) whose fibre $ F $ is homotopy equivalent to a compact CW-complex $ F $. The Becker–Gottlieb transfer is an $ S $- mapping $ \tau : B \rightarrow E $, which means that there is a mapping $ \tau : {\Sigma ^ {N} B } \rightarrow {\Sigma ^ {N} E } $ defined between the $ N $ th suspensions of $ B $ and $ E $ for some $ N $( cf. Suspension). Thus, $ \tau $ induces a homomorphism $ \tau _ {*} $ on any homology theory and $ \tau ^ {*} $ on any cohomology theory.

This gives striking relations on homology and cohomology, respectively:

$$ \tag{a1 } p _ {*} \circ \tau _ {*} = \chi ( F ) , \tau ^ {*} \circ p ^ {*} = \chi ( F ) , $$

where $ \chi ( F ) $ denotes multiplication by the Euler–Poincaré number of $ F $.

The Becker–Gottlieb transfer was discovered in the mid-1970s. It generalizes the transfer for finite covering spaces ( $ F $ is a finite set of points), which had been well-known since the 1940s and which was a generalization of a group theory transfer from the 1920s.

Since the discovery of the Becker–Gottlieb transfer, other transfers have been discovered which satisfy equations (a1) with the Euler–Poincaré number replaced by another elementary topological invariant. Thus,

$$ \tag{a2 } p _ {*} \circ \tau _ {*} = k, \quad \tau ^ {*} \circ p ^ {*} = k, $$

where $ k $ denotes multiplication by an integer $ k $, which can be a Lefschetz number, a coincidence number, a fixed-point index, a vector field index, or an intersection number. For any mapping $ p : X \rightarrow Y $, the greatest common divisor of all the integers $ k $ associated to a transfer is the Brouwer degree of $ p $ when $ X $ and $ Y $ are manifolds of the same dimension (cf. also Degree of a mapping). Some of these transfers are induced by $ S $- mappings, and others are not.

Several mathematicians contributed to the discovery of these other transfers, most of which either generalize some or all of the Becker–Gottlieb transfer. Dold's fixed-point transfer was discovered independently of the Becker–Gottlieb transfer, which it generalizes. It is sometimes called the Becker–Gottlieb–Dold transfer.

References

[a1] J.C. Becker, D.H. Gottlieb, "Vector fields and transfers" Manuscr. Math. , 72 (1991) pp. 111–130
How to Cite This Entry:
Becker-Gottlieb transfer. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Becker-Gottlieb_transfer&oldid=22073
This article was adapted from an original article by D.H. Gottlieb (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article