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A [[Spectrum of spaces|spectrum of spaces]], equivalent to the spectrum associated to a certain structure series (cf. [[B-Phi-structure|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t0926901.png" />-structure]]).
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t0926902.png" /> be a structure series, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t0926903.png" /> be the bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t0926904.png" /> induced by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t0926905.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t0926906.png" /> be the [[Thom space|Thom space]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t0926907.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t0926908.png" /> induces a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t0926909.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269010.png" /> is [[Suspension|suspension]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269012.png" /> is the one-dimensional trivial bundle). One obtains a spectrum of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269013.png" />, associated with the structure series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269014.png" />, and a Thom spectrum is any spectrum that is (homotopy) equivalent to a spectrum of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269015.png" />. It represents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269016.png" />-cobordism theory. Thus, the series of classical Lie groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269019.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269020.png" /> lead to the Thom spectra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269023.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269024.png" />.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269025.png" /> be Artin's braid group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269026.png" /> strings (cf. [[Braid theory|Braid theory]]). The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269028.png" /> is the symmetric group, yields a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269029.png" /> such that a structure series arises (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269030.png" /> is canonically imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269031.png" />). The corresponding Thom spectrum is equivalent to the Eilenberg–MacLane spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269032.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269033.png" /> is a Thom spectrum (cf. [[#References|[1]]], [[#References|[2]]]). Analogously, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269034.png" /> is a Thom spectrum, but using sphere bundles, [[#References|[3]]].
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A [[Spectrum of spaces|spectrum of spaces]], equivalent to the spectrum associated to a certain structure series (cf. [[B-Phi-structure| $  ( B, \phi ) $-
 +
structure]]).
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Let $  ( B _ {n} , \phi _ {n} , g _ {n} ) $
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be a structure series, and let  $  \xi _ {n} $
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be the bundle over  $  B _ {n} $
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induced by the mapping  $  \phi _ {n} :  B _ {n} \rightarrow  \mathop{\rm BO} _ {n} $.
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Let  $  T _ {n} $
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be the [[Thom space|Thom space]] of  $  \xi _ {n} $.
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The mapping  $  g _ {n} $
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induces a mapping  $  S _ {n} :  ST _ {n} \rightarrow T _ {n + 1 }  $,
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where  $  S $
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is [[Suspension|suspension]] and  $  ST \xi _ {n} = T( \xi _ {n} \oplus \theta ) $(
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$  \theta $
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is the one-dimensional trivial bundle). One obtains a spectrum of spaces  $  \{ T \xi _ {n} \} = T ( B, \phi , g) $,
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associated with the structure series  $  ( B _ {n} , \phi _ {n} , g _ {n} ) $,
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and a Thom spectrum is any spectrum that is (homotopy) equivalent to a spectrum of the form  $  T ( B, \phi , g) $.  
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It represents  $  ( B, \phi ) $-
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cobordism theory. Thus, the series of classical Lie groups  $  O _ {k} $,
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$  \mathop{\rm SO} _ {k} $,
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$  U _ {k} $,
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and  $  \mathop{\rm Sp} _ {k} $
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lead to the Thom spectra  $  \mathop{\rm TBO} $,
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$  \mathop{\rm TBSO} $,
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$  \mathop{\rm TBU} $,
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and  $  \mathop{\rm TBSp} $.
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Let  $  \beta _ {n} $
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be Artin's braid group on $  n $
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strings (cf. [[Braid theory|Braid theory]]). The homomorphism $  \beta _ {n} \rightarrow S _ {n} \subset  O _ {n} $,  
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where $  S _ {n} $
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is the symmetric group, yields a mapping $  B \beta _ {n} \rightarrow  \mathop{\rm BO} _ {n} $
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such that a structure series arises ( $  \beta _ {n} $
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is canonically imbedded in $  \beta _ {n + 1 }  $).  
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The corresponding Thom spectrum is equivalent to the Eilenberg–MacLane spectrum $  K ( \mathbf Z /2) = \{ K ( \mathbf Z /2, n) \} $,  
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so that $  K ( \mathbf Z /2) $
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is a Thom spectrum (cf. [[#References|[1]]], [[#References|[2]]]). Analogously, $  K ( \mathbf Z ) $
 +
is a Thom spectrum, but using sphere bundles, [[#References|[3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Mahowold,  "A new infinite family in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269035.png" />"  ''Topology'' , '''16'''  (1977)  pp. 249–256</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Priddy,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269036.png" /> as a Thom spectrum"  ''Proc. Amer. Math. Soc.'' , '''70''' :  2  (1978)  pp. 207–208</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Mahowold,  "Ring spectra which are Thom complexes"  ''Duke Math. J.'' , '''46''' :  3  (1979)  pp. 549–559</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Mahowold,  "A new infinite family in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269035.png" />"  ''Topology'' , '''16'''  (1977)  pp. 249–256</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Priddy,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092690/t09269036.png" /> as a Thom spectrum"  ''Proc. Amer. Math. Soc.'' , '''70''' :  2  (1978)  pp. 207–208</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Mahowold,  "Ring spectra which are Thom complexes"  ''Duke Math. J.'' , '''46''' :  3  (1979)  pp. 549–559</TD></TR></table>

Latest revision as of 08:25, 6 June 2020


A spectrum of spaces, equivalent to the spectrum associated to a certain structure series (cf. $ ( B, \phi ) $- structure).

Let $ ( B _ {n} , \phi _ {n} , g _ {n} ) $ be a structure series, and let $ \xi _ {n} $ be the bundle over $ B _ {n} $ induced by the mapping $ \phi _ {n} : B _ {n} \rightarrow \mathop{\rm BO} _ {n} $. Let $ T _ {n} $ be the Thom space of $ \xi _ {n} $. The mapping $ g _ {n} $ induces a mapping $ S _ {n} : ST _ {n} \rightarrow T _ {n + 1 } $, where $ S $ is suspension and $ ST \xi _ {n} = T( \xi _ {n} \oplus \theta ) $( $ \theta $ is the one-dimensional trivial bundle). One obtains a spectrum of spaces $ \{ T \xi _ {n} \} = T ( B, \phi , g) $, associated with the structure series $ ( B _ {n} , \phi _ {n} , g _ {n} ) $, and a Thom spectrum is any spectrum that is (homotopy) equivalent to a spectrum of the form $ T ( B, \phi , g) $. It represents $ ( B, \phi ) $- cobordism theory. Thus, the series of classical Lie groups $ O _ {k} $, $ \mathop{\rm SO} _ {k} $, $ U _ {k} $, and $ \mathop{\rm Sp} _ {k} $ lead to the Thom spectra $ \mathop{\rm TBO} $, $ \mathop{\rm TBSO} $, $ \mathop{\rm TBU} $, and $ \mathop{\rm TBSp} $.

Let $ \beta _ {n} $ be Artin's braid group on $ n $ strings (cf. Braid theory). The homomorphism $ \beta _ {n} \rightarrow S _ {n} \subset O _ {n} $, where $ S _ {n} $ is the symmetric group, yields a mapping $ B \beta _ {n} \rightarrow \mathop{\rm BO} _ {n} $ such that a structure series arises ( $ \beta _ {n} $ is canonically imbedded in $ \beta _ {n + 1 } $). The corresponding Thom spectrum is equivalent to the Eilenberg–MacLane spectrum $ K ( \mathbf Z /2) = \{ K ( \mathbf Z /2, n) \} $, so that $ K ( \mathbf Z /2) $ is a Thom spectrum (cf. [1], [2]). Analogously, $ K ( \mathbf Z ) $ is a Thom spectrum, but using sphere bundles, [3].

References

[1] M. Mahowold, "A new infinite family in " Topology , 16 (1977) pp. 249–256
[2] S. Priddy, " as a Thom spectrum" Proc. Amer. Math. Soc. , 70 : 2 (1978) pp. 207–208
[3] M. Mahowold, "Ring spectra which are Thom complexes" Duke Math. J. , 46 : 3 (1979) pp. 549–559
How to Cite This Entry:
Thom spectrum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thom_spectrum&oldid=48972
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article