View source for Sullivan conjecture

A conjecture in [[Homotopy|homotopy]] theory usually referring to a theorem about the contractibility, or homotopy equivalence, of certain types of mapping spaces. These results are vast generalizations of two different but related conjectures made by D. Sullivan in 1972.

H.T. Miller [[#References|[a1]]] achieved the first major breakthrough and is given credit for solving the Sullivan conjecture. This was published in 1984 and one version reads: The space of pointed mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s1203001.png" /> from the [[Classifying space|classifying space]] of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s1203002.png" /> to a finite [[CW-complex|CW-complex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s1203003.png" /> is weakly contractible. The mapping space has the compact-open topology.

An equivalent statement is that the space of unpointed mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s1203004.png" /> is weakly homotopy equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s1203005.png" /> (under the same hypotheses on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s1203006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s1203007.png" />).

These theorems are still true when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s1203008.png" /> is replaced by a CW-complex which has only finitely many non-zero homotopy groups, each of which is locally finite and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s1203009.png" /> can be any finite dimensional CW-complex. This improvement is due to A. Zabrodsky.

Equivariant versions of the Sullivan conjecture come about by considering the question: How close does the natural mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030010.png" /> come to being a homotopy equivalence? Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030011.png" /> is the fixed-point set of a group action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030012.png" /> on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030013.png" /> and the homotopy fixed-point set is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030014.png" />, the space of equivariant mappings from the contractible space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030015.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030016.png" /> acts freely to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030017.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030018.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030019.png" /> acting trivially on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030020.png" />, Miller's version of the Sullivan conjecture gives a positive answer to this question.

Another version of this question is that the fixed-point set of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030021.png" />-space localized at a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030022.png" /> is weakly homotopy equivalent to the homotopy fixed-point set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030023.png" /> acting on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030024.png" />-localization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030025.png" />. One proof of this result has been given by G. Carlsson, via the Segal conjecture [[#References|[a2]]]. Miller also independently proved this result, and J. Lannes has a subsequent proof using his <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030027.png" />-functor (cf. also [[Lannes-T-functor|Lannes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120300/s12030028.png" />-functor]]).

These theorems have found many beautiful applications at the hands of the above-mentioned mathematicians, as well as W.G. Dwyer, C. McGibbon, J.A. Neisendorfer and C. Wilkerson, and S. Jackowsky, to name only a few.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Miller,   "The Sullivan conjecture and homotopical representation theory" , ''Proc. Internat. Congress Math. (Berkeley, Calif., 1986)'' , '''1–2''' , Amer. Math. Soc.  (1987)  pp. 580–589</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Carlsson,   "Segal's Burnside ring conjecture and related problems in topology" , ''Proc. Internat. Congress Math. (Berkeley, Calif. 1986)'' , '''1–2''' , Amer. Math. Soc.  (1987)  pp. 574–579</TD></TR></table>