AbstractLet M be a smooth manifold and V a Euclidean space. Let $ \overline{{{\text{Emb}}}} $(M,V) be the homotopy fiber of the map Emb(M,V) → Imm(M,V). This paper is about the rational homology of $ \overline{{{\text{Emb}}}} $(M,V). We study it by applying embedding calculus and orthogonal calculus to the bifunctor (M,V)↦ HQ ∧ $ \overline{{{\text{Emb}}}} $(M,V)+. Our main theorem states that if $$ \dim V \geqslant 2{\text{ED}}{\left( M \right)} + 1 $$(where ED(M) is the embedding dimension of M), the Taylor tower in the sense of orthogonal calculus (henceforward called “the orthogonal tower”) of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at E1. In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich's theorem on the formality of the little balls operad. We write explicit formulas for the layers in the orthogonal tower of the functor $$ HQ \wedge \overline{{{\text{Emb}}}} {\left( {M,V} \right)}_{ + }. $$The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homotopy type of M. This, together with our rational splitting theorem, implies that, under the above assumption on codimension, rational homology equivalences of manifolds induce isomorphisms between the rational homology groups of $ \overline{{{\text{Emb}}}} $(–,V).

8 pages, 4 figures.-- PACS nrs.: 12.10.Dm; 12.10.Kt; 12.15.Ff; 14.60.Pq.-- ISI Article Identifier: 000247625400067.-- ArXiv pre-print available at: http://arxiv.org/abs/hep-ph/0607208

INTRODUCTION There are only a limited number of researches on the frequency, extent, causality and the location of injuries among young people. It is difficult to say to which extent the risky behavior in youth is really spread, because there are no routine data on this issue. In the Republic of Srpska, the first nationwide survey on health behavior of school aged children was conducted, comprising a very important area of health behavior related to injuries, violence and harassment. The aim of this paper was to investigate the risky behavior of school children in the Republic of Srpska in relation to injuries, violence and harassment. MATERIAL AND METHODS This cross-sectional study was conducted throughout the entire territory of the Republic of Srpska during 2002, on a selected sample of schools and covered a total of 1783 pupils, 15 years of age. The survey instrument was the international standard questionnaire, modified for the Republic of Srpska region. RESULTS Almost a quarter of all polled school children of both sexes have participated in fights. More than 10% of boys carry weapons. Injuries have mostly occurred during sports activities, on sports grounds (35.8%), at home (26.9%), in the school yard (14.5%), in the street or parking lots (11.4%) and in the countryside (5.6%). CONCLUSION The existing violence and injuries among school children indicate a clear need for improvement of mutual communication and tolerance among pupils and involvement of all relevant community members: parents, teachers, health workers and the entire society in health education of school children.