Spaces of sections of Banach algebra bundles

Suppose that $B$ is a $G$-Banach algebra over $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, $X$ is a finite dimensional compact metric space, $\zeta : P \to X$ is a standard principal $G$-bundle, and $A_\zeta = \Gamma (X, P \times_G B)$ is the associated algebra of sections. We produce a spectral sequence which converges to $\pi_*(GL_o A_\zeta)$ with [E^2_{-p,q} \cong \check{H}^p(X ; \pi_q(GL_o B)).] A related spectral sequence converging to \K_{*+1}(A_\zeta) (the real or complex topological $K$-theory) allows us to conclude that if $B$ is Bott-stable, (i.e., if \pi_*(GL_o B) \to \K_{*+1}(B) is an isomorphism for all $*>0$) then so is $A_\zeta$.Comment: 15 pages. Results generalized to include both real and complex K-theory. To appear in J. K-Theor

Brown representability for space-valued functors

In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify small contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every small contravariant functor from spaces to spaces which takes coproducts to products up to homotopy and takes homotopy pushouts to homotopy pullbacks is naturally weekly equivalent to a representable functor. The second representability theorem states: every contravariant continuous functor from the category of finite simplicial sets to simplicial sets taking homotopy pushouts to homotopy pullbacks is equivalent to the restriction of a representable functor. This theorem may be considered as a contravariant analog of Goodwillie's classification of linear functors.Comment: 19 pages, final version, accepted by the Israel Journal of Mathematic

A Note on the Non-Existence of Functors

We consider several cases of non-existence theorems for functors. For example, there are no nontrivial functors from the category of sets, (or the category of groups, or vector spaces) to any small category. See 2.3. Another kind of nonexistence is that of (co-)augmented functors. For example, every augmented functor from groups to abelian groups, is trivial, i.e. has a trivial augmentation map. Every surjective co-augmented functor from groups to perfect groups or to free groups is also trivial

PENERAPAN MODEL PEMBELAJARAN KOOPERATIF TIPE STAD DALAM PEMBELAJARAN SEJARAH UNTUK MENINGKATKAN KREATIVITAS DAN HASIL BELAJAR (Studi PTK pada Siswa Kelas XI di SMK Yasira Ciamis)

Tujuan dari penelitian ini adalah untuk menganalisis masalah penerapan pembelajaran sejarah menggunakan model pembelajaran kooperatif tipe Student Teams Achievement Divisions (STAD) untuk meningkatkan kreativitas dan hasil belajar pada siswa kelas XI di SMK Yasira Ciamis. Penelitian ini adalah penelitian tindakan kelas. Subjek penelitian ini adalah peserta didik kelas XI di SMK Yasira Ciamis dengan jumlah 24 peserta didik. Indikator yang diteliti dalam penelitian ini adalah kreativitas dan hasil belajar sejarah peserta didik. Hasil penelitian dari implementasi pembelajaran sejarah menggunakan model pembelajaran Kooperatif Tipe Student Teams Achievement Divisions (STAD) dapat meningkatkan kreativitas belajar siswa secara klasikal pada siklus I memperoleh sebesar 66,74% dengan kategori kurang kreatif, pada siklus II meningkat menjadi 82,26% dengan kategori cukup kreatif dan pada siklus III meningkat menjadi 83,65% dengan kategori kreatif. Pada implementasi pembelajaran sejarah menggunakan model pembelajaran kooperatif tipe STAD dapat meningkatkan hasil belajar, pada siklus I memperoleh persentase 45,83%, pada siklus II meningkat menjadi 79,17%, pada siklus III meningkat 91,67%. Hal ini menunjukkan bahwa pada tindakan siklus III ini lah dapat dikatakan berhasil walaupun masih ada 2 siswa yang belum tuntas, namun jika dilihat dari presentasi ketuntasan telah mencapai batas yang telah ditentukan yaitu sebesar 80% siswa yang tuntas

Cellular approximations and the Eilenberg-Moore spectral-sequence

We set up machinery for recognizing k-cellular modules and k-cellular approximations, where k is an R-module and R is either a ring or a ring-spectrum. Using this machinery we can identify the target of the Eilenberg-Moore cohomology spectral sequence for a fibration in various cases. In this manner we get new proofs for known results concerning the Eilenberg-Moore spectral sequence and generalize another result.Comment: 35 page

Cellular properties of nilpotent spaces

We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages, in particular classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield-Kan homology completion tower zkX whose terms we prove are all X–cellular for any X. As straightforward consequences, we show that if X is K–acyclic and nilpotent for a given homology theory K, then so are all its Postnikov sections PnX , and that any nilpotent space for which the space of pointed self-maps map .X; X/ is “canonically” discrete must be aspherical.Göran Gustafsson StiftelseFondo Europeo de Desarrollo RegionalMinisterio de Economía y Competitivida