The Nielsen Realization Problem for Non-Orientable Surfaces

We show the Teichm\"uller space of a non-orientable surface with marked points (considered as a Klein surface) can be identified with a subspace of the Teichm\"uller space of its orientable double cover. Also, it is well known that the mapping class group $\text{Mod} (N_g; k)$ of a non-orientable surface can be identified with a subgroup of $\text{Mod} (S_{g-1}; 2k)$, the mapping class group of its orientable double cover. These facts together with the classical Nielsen realization theorem are used to prove that every finite subgroup of $\text{Mod}(N_g; k)$ can be lifted isomorphically to a subgroup of the group of diffeomorphisms $\text{Diff}(N_g; k)$. In contrast, we show the projection $\text{Diff}(N_g) \to \text{Mod}(N_g)$ does not admit a section for large $g$.Comment: 17 pages, 1 figur

Configuration spaces and braid groups

In this snapshot we introduce configuration spaces and explain how a mathematician studies their ‘shape’. This will lead us to consider paths of configurations and braid groups, and to explore how algebraic properties of these groups determine features of the spaces

Periodicity of the pure mapping class group of non-orientable surfaces

We show that the pure mapping class group $\mathcal{N}_{g}^{k}$ of a non-orientable closed surface of genus $g\geqslant 2$ with $k\geqslant 1$ marked points has $p$-periodic cohomology for each odd prime $p$ for which $\mathcal{N}_{g}^{k}$ has $p$-torsion. Using the Yagita invariant and the cohomology classes obtained by the representation of subgroups of order $p$, we obtain that the $p$-period is less than or equal to $4$ when $g\geqslant 3$ and $k\geqslant 1$. Moreover, combining the Nielsen realization theorem and a characterization of the $p$-period given in terms of normalizers and centralizers of cyclic subgroups of order $p$, we show that the $p$-period of $\mathcal{N}_{g}^{k}$ is bounded below by $4$, whenever $\mathcal{N}_{g}^{k}$ has $p$-periodic cohomology, $g\geqslant 3$ and $k\geqslant 0$. These results provide partial answers to questions proposed by G. Hope and U. Tillmann.Comment: 16 page

Topological complexity of motion planning in projective product spaces

We study Farber's topological complexity (TC) of Davis' projective product spaces (PPS's). We show that, in many non-trivial instances, the TC of PPS's coming from at least two sphere factors is (much) lower than the dimension of the manifold. This is in high contrast with the known situation for (usual) real projective spaces for which, in fact, the Euclidean immersion dimension and TC are two facets of the same problem. Low TC-values have been observed for infinite families of non-simply connected spaces only for H-spaces, for finite complexes whose fundamental group has cohomological dimension not exceeding 2, and now in this work for infinite families of PPS's. We discuss general bounds for the TC (and the Lusternik-Schnirelmann category) of PPS's, and compute these invariants for specific families of such manifolds. Some of our methods involve the use of an equivariant version of TC. We also give a characterization of the Euclidean immersion dimension of PPS's through generalized concepts of axial maps and, alternatively, non-singular maps. This gives an explicit explanation of the known relationship between the generalized vector field problem and the Euclidean immersion problem for PPS's.Comment: 16 page

Production and purification of immunologically active core protein p24 from HIV-1 fused to ricin toxin B subunit in E. coli

<p>Abstract</p> <p>Background</p> <p>Gag protein from HIV-1 is a polyprotein of 55 kDa, which, during viral maturation, is cleaved to release matrix p17, core p24 and nucleocapsid proteins. The p24 antigen contains epitopes that prime helper CD4 T-cells, which have been demonstrated to be protective and it can elicit lymphocyte proliferation. Thus, p24 is likely to be an integral part of any multicomponent HIV vaccine. The availability of an optimal adjuvant and carrier to enhance antiviral responses may accelerate the development of a vaccine candidate against HIV. The aim of this study was to investigate the adjuvant-carrier properties of the B ricin subunit (RTB) when fused to p24.</p> <p>Results</p> <p>A fusion between ricin toxin B subunit and p24 HIV (RTB/p24) was expressed in <it>E. coli</it>. Affinity chromatography was used for purification of p24 alone and RTB/p24 from cytosolic fractions. Biological activity of RTB/p24 was determined by ELISA and affinity chromatography using the artificial receptor glycoprotein asialofetuin. Both assays have demonstrated that RTB/p24 is able to interact with complex sugars, suggesting that the chimeric protein retains lectin activity. Also, RTB/p24 was demonstrated to be immunologically active in mice. Two weeks after intraperitoneal inoculation with RTB/p24 without an adjuvant, a strong anti-p24 immune response was detected. The levels of the antibodies were comparable to those found in mice immunized with p24 alone in the presence of Freund adjuvant. RTB/p24 inoculated intranasally in mice, also elicited significant immune responses to p24, although the response was not as strong as that obtained in mice immunized with p24 in the presence of the mucosal adjuvant cholera toxin.</p> <p>Conclusion</p> <p>In this work, we report the expression in <it>E. coli </it>of HIV-1 p24 fused to the subunit B of ricin toxin. The high levels of antibodies obtained after intranasal and intraperitoneal immunization of mice demonstrate the adjuvant-carrier properties of RTB when conjugated to an HIV structural protein. This is the first report in which a eukaryotic toxin produced in <it>E. coli </it>is employed as an adjuvant to elicit immune responses to p24 HIV core antigen.</p

Product decomposition of loop spaces of configuration spaces

AbstractThe configuration space of k points in RPn, CPn and HPn are studied. In this article we show that after looping once, they split as a product of spheres and the loop space of certain orbit configuration spaces

Orbit configuration spaces associated to the Gaussian integers: homotopy and homology groups

AbstractThe purpose of this article is to analyze several Lie algebras associated to “orbit configuration spaces” obtained from the standard integral lattice Z+iZ in the complex numbers. The Lie algebra obtained from the descending central series for the associated fundamental group is shown to be isomorphic, up to a regrading, to the Lie algebra obtained from the higher homotopy groups of “higher dimensional arrangements” modulo torsion. The resulting Lie algebras are similar to those studied by T. Kohno associated to elliptic KZ systems [Topology Appl. 78 (1997) 79–94]. A question about the generality of this behavior is posed