Duality between stable strong shape morphisms and stable homotopy classes
Let SStrShn be the full subcategory of the stable strong shape category SStrSh of pointed compacta [H-N] whose objects are all pointed subcompacta of S and let SOn be the full subcategory of the stable homotopy category S ([S-W] or [S]) whose objects are all open subsets of S. In this paper it is shown that there exists a contravariant additive functor Dn : SStrShn → SOn such that Dn(X) = S \ X for every subcompactum X of S and Dn : SStrShn(X, Y ) → SOn(S \Y,S \X) is an isomorphism of abelian groups for all compacta X,Y ⊂ S. Moreover, if X ⊂ Y ⊂ S, j : S \ Y → S \ X is an inclusion and α ∈ SStrShn(X, Y ) is induced by the inclusion of X into Y then Dn(α) = {j}. Introduction. Basic definitions. In [N] it has been proved that the stable shape category of subcompacta of S is isomorphic to the stable weak homotopy category of their complements. This theorem generalizes the Spanier-Whitehead Duality Theorem and corresponds to the Chapman Complement Theorem [S1]. In [H-N] the authors have constructed a stable strong shape category of pointed metric compacta (see also [M]). The purpose of the present note is to prove that there exists a contravariant functor from the stable strong shape category of pointed subcompacta of 2000 Mathematics Subject Classification. 55P55, 55P25.
A strong-shape theoretical version of a result due to E. Lima
Abstract There exists a strong-shape theoretical version of S-duality for two compacta X , Y ⊂ S n (Theorem 1.1) which turns out to be an immediate consequence of Alexander duality for generalized (co-)homology theories Ē ∗ , Ē ∗ with coefficients in a spectrum E , as in [1]. This result transfers earlier investigations by Lima [4] from ordinary to strong shape theory.
Dawn or dusk of the 5th age of research in educational technology? A literature review on (e-)leadership for technology-enhanced learning in higher education (2013-2017)
The aim of this article is to establish the extent to which the concept of e-leadership has taken off as a lens through which to study leadership for technology-enhanced learning (TEL) in higher education. Building on a previous study conducted in 2013, this article thus covers an exploratory review of the literature for the period 2103-2017. It analyses 49 articles which explore both the specific concept of e-leadership as well as other work dealing more generally with leadership and organisational change for TEL in higher education. The findings show that while none of the empirical studies identified in the literature refer explicitly to e-leadership, there are a number of interesting insights to be found in the theoretical articles. The results also highlight the widely different interpretations and applications of the concept of e-leadership and the consequent need for the definition to be refined. The paper concludes with recommendations for further multidisciplinary research at the intersection of the fields of educational technology and educational management, focusing on values, strategy, organisation and leadership interactions at meso level, the economy and public policy at macro level, and teaching and learning at the micro level, as well as for research in Leadership Development for TEL.