View source for Federer spectral sequence

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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} $-
stage consists of the singular cohomology (cf. [[Singular homology|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|Connected space]]) and  $  u : X \rightarrow Y $
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} ) $, 
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 [[#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.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Federer,   "A study of function spaces by spectral sequences"  ''Trans. Amer. Math. Soc.'' , '''82'''  (1956)  pp. 340–361</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.G. Barratt,   "Track groups I, II"  ''Proc. London Math. Soc.'' , '''5'''  (1955)  pp. 71–106; 285–329</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Brown,   "On Künneth suspensions"  ''Proc. Cambridge. Philos. Soc.'' , '''60'''  (1964)  pp. 713–720</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.M. Møller,   "On equivariant function spaces"  ''Pacific J. Math.'' , '''142'''  (1990)  pp. 103–119</TD></TR></table>