View source for Federer spectral sequence

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.

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

<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>

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" />.

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>