Characterizations of spectra

with -injective cohomology which satisfy

the Brown-Gitler property

Authors:
David J. Hunter and Nicholas J. Kuhn

Journal:
Trans. Amer. Math. Soc. **352** (2000), 1171-1190

MSC (1991):
Primary 55P42; Secondary 55T15, 55T20

DOI:
https://doi.org/10.1090/S0002-9947-99-02375-2

Published electronically:
February 15, 1999

MathSciNet review:
1621749

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Abstract | References | Similar Articles | Additional Information

Abstract: We work in the stable homotopy category of -complete connective spectra having mod homology of finite type. means cohomology with coefficients, and is a left module over the Steenrod algebra .

A spectrum is called *spacelike* if it is a wedge summand of a suspension spectrum, and a spectrum *satisfies the Brown-Gitler property* if the natural map is onto, for all spacelike .

It is known that there exist spectra satisfying the Brown-Gitler property, and with isomorphic to the injective envelope of in the category of unstable -modules.

Call a spectrum *standard* if it is a wedge of spectra of the form , where is a stable wedge summand of the classifying space of some elementary abelian -group. Such spectra have -injective cohomology, and all -injectives appear in this way.

Working directly with the two properties of stated above, we clarify and extend earlier work by many people on Brown-Gitler spectra. Our main theorem is that, if is a spectrum with -injective cohomology, the following conditions are equivalent:

(A) there exist a spectrum whose cohomology is a reduced -injective and a map that is epic in cohomology, (B) there exist a spacelike spectrum and a map that is epic in cohomology, (C) is monic in cohomology, (D) satisfies the Brown-Gitler property, (E) is spacelike, (F) is standard. ( is *reduced* if it has no nontrivial submodule which is a suspension.)

As an application, we prove that the Snaith summands of are Brown-Gitler spectra-a new result for the most interesting summands at odd primes. Another application combines the theorem with the second author's work on the Whitehead conjecture.

Of independent interest, enroute to proving that (B) implies (C), we prove that the homology suspension has the following property: if an -connected space admits a map to an -fold suspension that is monic in mod homology, then is onto in mod homology.

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Additional Information

**David J. Hunter**

Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903

Address at time of publication:
Department of Mathematics, North Central College, Naperville, Illinois 60540

Email:
dahunter@noctrl.edu

**Nicholas J. Kuhn**

Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903

Email:
njk4x@virginia.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02375-2

Received by editor(s):
August 27, 1997

Published electronically:
February 15, 1999

Additional Notes:
Research by the second author was partially supported by the N.S.F

Article copyright:
© Copyright 1999
American Mathematical Society