A classification of smooth embeddings of 3-manifolds in 6-space

We work in the smooth category. If there are knotted embeddings S^n\to R^m, which often happens for 2m<3n+4, then no concrete complete description of embeddings of n-manifolds into R^m up to isotopy was known, except for disjoint unions of spheres. Let N be a closed connected orientable 3-manifold. Our main result is the following description of the set Emb^6(N) of embeddings N\to R^6 up to isotopy. The Whitney invariant W : Emb^6(N) \to H_1(N;Z) is surjective. For each u \in H_1(N;Z) the Kreck invariant \eta_u : W^{-1}u \to Z_{d(u)} is bijective, where d(u) is the divisibility of the projection of u to the free part of H_1(N;Z). The group Emb^6(S^3) is isomorphic to Z (Haefliger). This group acts on Emb^6(N) by embedded connected sum. It was proved that the orbit space of this action maps under W bijectively to H_1(N;Z) (by Vrabec and Haefliger's smoothing theory). The new part of our classification result is determination of the orbits of the action. E. g. for N=RP^3 the action is free, while for N=S^1\times S^2 we construct explicitly an embedding f : N \to R^6 such that for each knot l:S^3\to R^6 the embedding f#l is isotopic to f. Our proof uses new approaches involving the Kreck modified surgery theory or the Boechat-Haefliger formula for smoothing obstruction.Comment: 32 pages, a link to http://www.springerlink.com added, to appear in Math. Zei

Spontaneous Atopic Dermatitis-Like Symptoms in a/a ma ft/ma ft/J Flaky Tail Mice Appear Early after Birth.

Loss-of-function mutations in human profilaggrin gene have been identified as the cause of ichthyosis vulgaris (IV), and as a major predisposition factor for atopic dermatitis (AD). Similarly, flaky tail (a/a ma ft/ma ft/J) mice were described as a model for IV, and shown to be predisposed to eczema. The aim of this study was to correlate the flaky tail mouse phenotype with human IV and AD, in order to dissect early molecular events leading to atopic dermatitis in mice and men, suffering from filaggrin deficiency. Thus, 5-days old flaky tail pups were analyzed histologically, expression of cytokines was measured in skin and signaling pathways were investigated by protein analysis. Human biopsies of IV and AD patients were analyzed histologically and by real time PCR assays. Our data show acanthosis and hyperproliferation in flaky tail epidermis, associated with increased IL1β and thymic stromal lymphopoietin (TSLP) expression, and Th2-polarization. Consequently, NFκB and Stat pathways were activated, and IL6 mRNA levels were increased. Further, quantitative analysis of late epidermal differentiation markers revealed increased Small proline-rich protein 2A (Sprr2a) synthesis. Th2-polarization and Sprr2a increase may result from high TSLP expression, as shown after analysis of 5-days old K14-TSLP tg mouse skin biopsies. Our findings in the flaky tail mouse correlate with data obtained from patient biopsies of AD, but not IV. We propose that proinflammatory cytokines are responsible for acanthosis in flaky tail epidermis, and together with the Th2-derived cytokines lead to morphological changes. Accordingly, the a/a ma ft/ma ft/J mouse model can be used as an appropriate model to study early AD onset associated with profilaggrin deficiency