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The Federer spectral sequence is a means to compute homotopy groups of function spaces (track groups; cf. [[Homotopy group|Homotopy group]]). Its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110030/f1100301.png" />-stage consists of the singular cohomology (cf. [[Singular homology|Singular homology]]) of the source with coefficients in the homotopy groups of the target.
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More specifically, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110030/f1100302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110030/f1100303.png" /> be connected topological spaces (cf. [[Connected space|Connected space]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110030/f1100304.png" /> a [[Continuous mapping|continuous mapping]]. The Federer spectral sequence for this situation is a second quadrant homology [[Spectral sequence|spectral sequence]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110030/f1100305.png" />, with
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<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/f/f110/f110030/f1100306.png" /></td> </tr></table>
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The Federer spectral sequence is a means to compute homotopy groups of function spaces (track groups; cf. [[Homotopy group|Homotopy group]]). Its  $  E _ {2} $-
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stage consists of the singular cohomology (cf. [[Singular homology|Singular homology]]) of the source with coefficients in the homotopy groups of the target.
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110030/f1100307.png" /> otherwise. Under appropriate finiteness conditions it converges to the homotopy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110030/f1100308.png" /> of the space of continuous mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110030/f1100309.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110030/f11003010.png" />.
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More specifically, let  $  X $
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and $  Y $
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be connected topological spaces (cf. [[Connected space|Connected space]]) and  $  u : X \rightarrow Y $
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a [[Continuous mapping|continuous mapping]]. The Federer spectral sequence for this situation is a second quadrant homology [[Spectral sequence|spectral sequence]]  $  ( E _ {** }  ^ {r} ,d  ^ {r} ) $,
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with
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$$
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E _ {pq }  ^ {2} = H ^ {- p } ( X; \pi _ {q} ( Y ) )  \textrm{ for  }  p + q \geq  0
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$$
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and  $  E _ {pq }  ^ {2} = 0 $
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otherwise. Under appropriate finiteness conditions it converges to the homotopy group $  \pi _ {p + q }  ( { \mathop{\rm map} } ( X,Y ) ,u ) $
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of the space of continuous mappings from $  X $
 +
to $  Y $.
  
 
In the literature, this spectral sequence occurs explicitly in [[#References|[a1]]] and implicitly in [[#References|[a2]]] (and is for this reason sometimes referred to as the Barratt–Federer spectral sequence). See [[#References|[a3]]] or [[#References|[a4]]] for later generalizations and modifications.
 
In the literature, this spectral sequence occurs explicitly in [[#References|[a1]]] and implicitly in [[#References|[a2]]] (and is for this reason sometimes referred to as the Barratt–Federer spectral sequence). See [[#References|[a3]]] or [[#References|[a4]]] for later generalizations and modifications.

Latest revision as of 19:38, 5 June 2020


The Federer spectral sequence is a means to compute homotopy groups of function spaces (track groups; cf. Homotopy group). Its $ E _ {2} $- stage consists of the singular cohomology (cf. Singular homology) of the source with coefficients in the homotopy groups of the target.

More specifically, let $ X $ and $ Y $ be connected topological spaces (cf. Connected space) and $ u : X \rightarrow Y $ a continuous mapping. The Federer spectral sequence for this situation is a second quadrant homology spectral sequence $ ( E _ {** } ^ {r} ,d ^ {r} ) $, with

$$ E _ {pq } ^ {2} = H ^ {- p } ( X; \pi _ {q} ( Y ) ) \textrm{ for } p + q \geq 0 $$

and $ E _ {pq } ^ {2} = 0 $ otherwise. Under appropriate finiteness conditions it converges to the homotopy group $ \pi _ {p + q } ( { \mathop{\rm map} } ( X,Y ) ,u ) $ of the space of continuous mappings from $ X $ to $ Y $.

In the literature, this spectral sequence occurs explicitly in [a1] and implicitly in [a2] (and is for this reason sometimes referred to as the Barratt–Federer spectral sequence). See [a3] or [a4] for later generalizations and modifications.

References

[a1] H. Federer, "A study of function spaces by spectral sequences" Trans. Amer. Math. Soc. , 82 (1956) pp. 340–361
[a2] M.G. Barratt, "Track groups I, II" Proc. London Math. Soc. , 5 (1955) pp. 71–106; 285–329
[a3] R. Brown, "On Künneth suspensions" Proc. Cambridge. Philos. Soc. , 60 (1964) pp. 713–720
[a4] J.M. Møller, "On equivariant function spaces" Pacific J. Math. , 142 (1990) pp. 103–119
How to Cite This Entry:
Federer spectral sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Federer_spectral_sequence&oldid=46909
This article was adapted from an original article by J.M. Møller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article