Towers and injective cohomology algebras

Goerss, Paul; Smith, Larry

doi:10.1090/S0002-9947-1987-0902789-X

Towers and injective cohomology algebras
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by Paul Goerss and Larry Smith PDF

Trans. Amer. Math. Soc. 303 (1987), 619-636 Request permission

Abstract:

Let $Y$ be a space of finite type such that ${H^{\ast }}Y$ is injective as an unstable algebra over the Steenrod algebra $A$ and such that ${\overline H ^{\ast }}Y$ is $A$-unbounded. Let $X$ be a simply connected $p$-complete space. Then any map of $A$-algebras $f:{H^{\ast }}\Omega X \to {H^{\ast }}Y$ can be realized as a map of spaces.

References

Jaume Aguadé and Larry Smith, Modular cohomology algebras, Amer. J. Math. 107 (1985), no. 3, 507–530. MR 789653, DOI 10.2307/2374367
Edgar H. Brown Jr. and Samuel Gitler, A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra, Topology 12 (1973), 283–295. MR 391071, DOI 10.1016/0040-9383(73)90014-1
E. H. Brown Jr. and F. P. Peterson, On the stable decomposition of $\Omega ^{2}S^{r+2}$, Trans. Amer. Math. Soc. 243 (1978), 287–298. MR 500933, DOI 10.1090/S0002-9947-1978-0500933-4
A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573
Gunnar Carlsson, G. B. Segal’s Burnside ring conjecture for $(\textbf {Z}/2)^{k}$, Topology 22 (1983), no. 1, 83–103. MR 682060, DOI 10.1016/0040-9383(83)90046-0
Paul G. Goerss, A direct construction for the duals of Brown-Gitler spectra, Indiana Univ. Math. J. 34 (1985), no. 4, 733–751. MR 808823, DOI 10.1512/iumj.1985.34.34039
Paul G. Goerss, Unstable projectives and stable $\textrm {Ext}$: with applications, Proc. London Math. Soc. (3) 53 (1986), no. 3, 539–561. MR 868458, DOI 10.1112/plms/s3-53.3.539
Jean Lannes and Saïd Zarati, Sur les foncteurs dérivés de la déstabilisation, Math. Z. 194 (1987), no. 1, 25–59 (French). MR 871217, DOI 10.1007/BF01168004
Jean Lannes and Saïd Zarati, Sur les ${\scr U}$-injectifs, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 2, 303–333 (French, with English summary). MR 868302
W. S. Massey and F. P. Peterson, The cohomology structure of certain fibre spaces. I, Topology 4 (1965), 47–65. MR 189032, DOI 10.1016/0040-9383(65)90048-0

The

cohomology structure of certain fibre spaces

Haynes Miller, Massey-Peterson towers and maps from classifying spaces, Algebraic topology, Aarhus 1982 (Aarhus, 1982) Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 401–417. MR 764593, DOI 10.1007/BFb0075581
Haynes Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. (2) 120 (1984), no. 1, 39–87. MR 750716, DOI 10.2307/2007071
John C. Moore and Larry Smith, Hopf algebras and multiplicative fibrations. I, Amer. J. Math. 90 (1968), 752–780. MR 234455, DOI 10.2307/2373482
Larry Smith, Realizing certain polynomial algebras as cohomology rings of spaces of finite type fibered over $\times B\textrm {U}(d)$, Pacific J. Math. 126 (1987), no. 2, 361–377. MR 869783
Larry Smith and R. E. Stong, On the invariant theory of finite groups: orbit polynomials and splitting principles, J. Algebra 110 (1987), no. 1, 134–157. MR 904185, DOI 10.1016/0021-8693(87)90040-8
Robert J. Wellington, The unstable Adams spectral sequence for free iterated loop spaces, Mem. Amer. Math. Soc. 36 (1982), no. 258, viii+225. MR 646741, DOI 10.1090/memo/0258
Jean Lannes and Lionel Schwartz, À propos de conjectures de Serre et Sullivan, Invent. Math. 83 (1986), no. 3, 593–603 (French, with English summary). MR 827370, DOI 10.1007/BF01394425

Sur la cohomologie des

groupes elementaire