A note on Sullivan completion
Yasumasa Hirashima · Osaka City University (Osaka City University) · 1980
In this note we give an alternative construction of the Sullivan finite completion for a "good" space by only making use of the scandard techniques in homotopy theory.Let c: X->Y be a based map of connected based spaces.We say that c is a TΓϊjc-finite completion of X if it is the finite completion on n x and πΊ-fϊnite completion of the higher homotopy.Theorem 0 (Sullivan [8, Theorem 3.1.ii), Corollary of proof]). Let X be a connected based space with "good" homotopy groups. A map c: X->Y is equivalent to the finite completion if and only if c is a π*-finite completion.Sullivan [8; Theorem 3.1.i)] also shows that sufficiently many spaces have "good" homotopy groups.Thus, to construct Sullivan finite completion, it is enough to construct a Tr^-finite completion.Since our arguments are quite formal, analogous /-finite construction is also available for a set / of primes.