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''de Rham torsion, Franz torsion''
 
''de Rham torsion, Franz torsion''
  
An invariant which allows one to distinguish many structures in differential topology, for example knots and smooth structures on manifolds, particularly on lens spaces. Reidemeister torsion was first introduced by K. Reidemeister (see [[#References|[1]]]) while studying three-dimensional lenses, the generalization for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r0809601.png" />-dimensional lenses was obtained independently in [[#References|[2]]] and [[#References|[3]]].
+
An invariant which allows one to distinguish many structures in differential topology, for example knots and smooth structures on manifolds, particularly on lens spaces. Reidemeister torsion was first introduced by K. Reidemeister (see [[#References|[1]]]) while studying three-dimensional lenses, the generalization for $  n $-
 +
dimensional lenses was obtained independently in [[#References|[2]]] and [[#References|[3]]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r0809602.png" /> be a free complex of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r0809603.png" />-modules, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r0809604.png" /> is an associative ring with a unit element. Further, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r0809605.png" /> be a matrix representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r0809606.png" />, i.e. a homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r0809607.png" /> into the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r0809608.png" /> of all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r0809609.png" />-matrices. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096010.png" /> be distinguished bases in the modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096011.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096012.png" />, and let the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096014.png" />-modules be acyclic; then the [[Whitehead torsion|Whitehead torsion]] is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096016.png" /> is the multiplicative group of the field of real numbers. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096017.png" /> is called the Reidemeister torsion of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096018.png" />, and also the real Reidemeister torsion.
+
Let $  C $
 +
be a free complex of left $  A $-
 +
modules, where $  A $
 +
is an associative ring with a unit element. Further, let $  h $
 +
be a matrix representation of $  A $,  
 +
i.e. a homomorphism from $  A $
 +
into the ring $  \mathbf R ^ {n \times n } $
 +
of all real $  ( n \times n) $-
 +
matrices. Let $  c _ {k} $
 +
be distinguished bases in the modules $  C _ {k} $
 +
of the complex $  C $,  
 +
and let the complex $  C  ^  \prime  = \mathbf R ^ {n \times n } \otimes _ {A} C $
 +
of $  \mathbf R ^ {n \times n } $-
 +
modules be acyclic; then the [[Whitehead torsion|Whitehead torsion]] is defined as $  \tau ( C  ^  \prime  ) \in \overline{K}\; _ {1} \mathbf R ^ {n \times n } = \overline{K}\; _ {1} \mathbf R = \mathbf R _ {+} $,  
 +
where $  \mathbf R _ {+} $
 +
is the multiplicative group of the field of real numbers. The number $  \tau ( C  ^  \prime  ) $
 +
is called the Reidemeister torsion of the complex $  C  ^  \prime  $,  
 +
and also the real Reidemeister torsion.
  
The usefulness of transforming the Whitehead torsion into the Reidemeister torsion is based on Bass' theorem [[#References|[4]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096019.png" /> is a finite group, then the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096020.png" /> has finite order if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096021.png" /> for any representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096023.png" /> is the Reidemeister torsion induced by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096024.png" />.
+
The usefulness of transforming the Whitehead torsion into the Reidemeister torsion is based on Bass' theorem [[#References|[4]]]. If $  \pi $
 +
is a finite group, then the element $  \omega \in  \mathop{\rm Wh} ( \pi ) $
 +
has finite order if $  h _ {*} ( \omega ) = 1 $
 +
for any representation $  h $,  
 +
where $  h _ {*} ( \omega ) $
 +
is the Reidemeister torsion induced by the element $  \omega $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Reidemeister,  "Homotopieringe und Linsenräume"  ''Abh. Math. Sem. Univ. Hamburg'' , '''11'''  (1935)  pp. 102–109</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Franz,  "Ueber die Torsion einer Ueberdeckung"  ''J. Reine Angew. Math.'' , '''173'''  (1935)  pp. 245–254</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. de Rham,  "Sur les nouveaux invariants de M. Reidemeister"  ''Mat. Sb.'' , '''1''' :  5  (1936)  pp. 737–743</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Bass,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096025.png" />-theory and stable algebra"  ''Publ. Math. IHES'' , '''22'''  (1964)  pp. 5–60</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Reidemeister,  "Homotopieringe und Linsenräume"  ''Abh. Math. Sem. Univ. Hamburg'' , '''11'''  (1935)  pp. 102–109</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Franz,  "Ueber die Torsion einer Ueberdeckung"  ''J. Reine Angew. Math.'' , '''173'''  (1935)  pp. 245–254</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. de Rham,  "Sur les nouveaux invariants de M. Reidemeister"  ''Mat. Sb.'' , '''1''' :  5  (1936)  pp. 737–743</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Bass,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080960/r08096025.png" />-theory and stable algebra"  ''Publ. Math. IHES'' , '''22'''  (1964)  pp. 5–60</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Milnor,  "Whitehead torsion"  ''Bull. Amer. Math. Soc.'' , '''72'''  (1966)  pp. 358–426</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Milnor,  "Whitehead torsion"  ''Bull. Amer. Math. Soc.'' , '''72'''  (1966)  pp. 358–426</TD></TR></table>

Latest revision as of 08:10, 6 June 2020


de Rham torsion, Franz torsion

An invariant which allows one to distinguish many structures in differential topology, for example knots and smooth structures on manifolds, particularly on lens spaces. Reidemeister torsion was first introduced by K. Reidemeister (see [1]) while studying three-dimensional lenses, the generalization for $ n $- dimensional lenses was obtained independently in [2] and [3].

Let $ C $ be a free complex of left $ A $- modules, where $ A $ is an associative ring with a unit element. Further, let $ h $ be a matrix representation of $ A $, i.e. a homomorphism from $ A $ into the ring $ \mathbf R ^ {n \times n } $ of all real $ ( n \times n) $- matrices. Let $ c _ {k} $ be distinguished bases in the modules $ C _ {k} $ of the complex $ C $, and let the complex $ C ^ \prime = \mathbf R ^ {n \times n } \otimes _ {A} C $ of $ \mathbf R ^ {n \times n } $- modules be acyclic; then the Whitehead torsion is defined as $ \tau ( C ^ \prime ) \in \overline{K}\; _ {1} \mathbf R ^ {n \times n } = \overline{K}\; _ {1} \mathbf R = \mathbf R _ {+} $, where $ \mathbf R _ {+} $ is the multiplicative group of the field of real numbers. The number $ \tau ( C ^ \prime ) $ is called the Reidemeister torsion of the complex $ C ^ \prime $, and also the real Reidemeister torsion.

The usefulness of transforming the Whitehead torsion into the Reidemeister torsion is based on Bass' theorem [4]. If $ \pi $ is a finite group, then the element $ \omega \in \mathop{\rm Wh} ( \pi ) $ has finite order if $ h _ {*} ( \omega ) = 1 $ for any representation $ h $, where $ h _ {*} ( \omega ) $ is the Reidemeister torsion induced by the element $ \omega $.

References

[1] K. Reidemeister, "Homotopieringe und Linsenräume" Abh. Math. Sem. Univ. Hamburg , 11 (1935) pp. 102–109
[2] W. Franz, "Ueber die Torsion einer Ueberdeckung" J. Reine Angew. Math. , 173 (1935) pp. 245–254
[3] G. de Rham, "Sur les nouveaux invariants de M. Reidemeister" Mat. Sb. , 1 : 5 (1936) pp. 737–743
[4] H. Bass, "-theory and stable algebra" Publ. Math. IHES , 22 (1964) pp. 5–60

Comments

References

[a1] J. Milnor, "Whitehead torsion" Bull. Amer. Math. Soc. , 72 (1966) pp. 358–426
How to Cite This Entry:
Reidemeister torsion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reidemeister_torsion&oldid=48494
This article was adapted from an original article by A.S. Mishchenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article