Contents xiii §23.3. Using the Diagonal Map to Construct Cohomology Operations §23.4. The Steenrod Reduced Powers §23.5. TheÁdem Relations §23.6. The Algebra of the Steenrod Algebra §23.7. Wrap-Up Chapter 24. Chain Complexes §24.1. The Cellular Complex §24.2. Applying Algebraic Universal Coefficients Theorems §24.3. The General Künneth Theorem §24.4. Algebra Structures on C * (X) and C * (X) §24.5. The Singular Chain Complex Chapter 25. Topics, Problems and Projects §25.1. Algebra Structures on R n and C n §25.2. Relative Cup Products §25.3. Hopf Invariants and Hopf Maps §25.4. Some Homotopy Groups of Spheres §25.5. The Borsuk-Ulam Theorem §25.6. Moore Spaces and Homology Decompositions §25.7. Finite Generation of π * (X) and H * (X) §25.8. Surfaces §25.9. Euler Characteristic §25.10. The Künneth Theorem via Symmetric Products §25.11. The Homology Algebra of ΩΣX §25.12. The Adjoint λ X of id ΩX §25.13. Some Algebraic Topology of Fibrations §25.14. A Glimpse of Spectra §25.15. A Variety of Topics §25.16.

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Abstract. We show that the Lusternik-Schnirelmann category of the symplectic group Sp(3) is 5. This L-S category coincides with the cone length and the stable weak category.

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Abstract. Let A be a fixed collection of spaces, and suppose K is a nilpotent space that can be built from spaces in A by a succession of cofiber sequences. We show that, under mild conditions on the collection A, it is possible to construct K from spaces in A using, instead, homotopy (inverse) limits and extensions by fibrations. One consequence is that if K is a nilpotent finite complex, then ΩK can be built from finite wedges of spheres using homotopy limits and extensions by fibrations. This is applied to show that if map * (X, S n ) is weakly contractible for all sufficiently large n, then map * (X, K) is weakly contractible for any nilpotent finite complex K. Introduction.A Miller space is a CW complex X with the property that the space map * (X, K) of pointed maps from X to K is weakly contractible for any nilpotent finite complex K (cf. [8, p. 46]). They are named for Haynes Miller, who proved in his landmark paper [16] that if G is a (locally) finite group, then the classifying space BG is a Miller space. In this paper we prove the following simple recognition principle for Miller spaces.Theorem (Corollary 11). Let X be a space and let N ∈ N. Then the following are equivalent: