## Towers and injective cohomology algebras

HTML articles powered by AMS MathViewer

- 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
—, - 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
J. Lannes,

*The*$\bmod 2$

*cohomology structure of certain fibre spaces*, Mem. Amer. Math. Soc. No. 74, 1967.

*Sur la cohomologie des*$p$-

*groupes elementaire*, preprint, École Polytechnique, 1986.

## Additional Information

- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**303**(1987), 619-636 - MSC: Primary 55P15; Secondary 55S35
- DOI: https://doi.org/10.1090/S0002-9947-1987-0902789-X
- MathSciNet review: 902789