Federer spectral sequence

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