The orientation-preserving diffeomorphism group of S^2 deforms to SO(3) smoothly

Smale proved that the orientation-preserving diffeomorphism group of S^2 has a continuous strong deformation retraction to SO(3). In this paper, we construct such a strong deformation retraction which is diffeologically smooth.Comment: 16 page

Control of germ-band retraction in Drosophila by the zinc-finger protein HINDSIGHT

Drosophila embryos lacking hindsight gene function have a normal body plan and undergo normal germ-band extension. However, they fail to retract their germ bands. hindsight encodes a large nuclear protein of 1920 amino acids that contains fourteen C2H2-type zinc fingers, and glutamine-rich and proline-rich domains, suggesting that it functions as a transcription factor. Initial embryonic expression of hindsight RNA and protein occurs in the endoderm (midgut) and extraembryonic membrane (amnioserosa) prior to germ-band extension and continues in these tissues beyond the completion of germ-band retraction. Expression also occurs in the developing tracheal system, central and peripheral nervous systems, and the ureter of the Malpighian tubules. Strikingly, hindsight is not expressed in the epidermal ectoderm which is the tissue that undergoes the cell shape changes and movements during germ-band retraction. The embryonic midgut can be eliminated without affecting germ-band retraction. However, elimination of the amnioserosa results in the failure of germ-band retraction, implicating amnioserosal expression of hindsight as crucial for this process. Ubiquitous expression of hindsight in the early embryo rescues germ-band retraction without producing dominant gainof-function defects, suggesting that hindsight’s role in germ-band retraction is permissive rather than instructive. Previous analyses have shown that hindsight is required for maintenance of the differentiated amnioserosa (Frank, L. C. and Rushlow, C. (1996) Development 122, 1343-1352). Two classes of models are consistent with the present data. First, hindsight’s function in germ-band retraction may be limited to maintenance of the amnioserosa which then plays a physical role in the retraction process through contact with cells of the epidermal ectoderm. Second, hindsight might function both to maintain the amnioserosa and to regulate chemical signaling from the amnioserosa to the epidermal ectoderm, thus coordinating the cell shape changes and movements that drive germ-band retraction

Simultaneously continuous retraction and Bishop-Phelps-Bollob\'as type theorem

We study the existence of a retraction from the dual space $X^*$ of a (real or complex) Banach space $X$ onto its unit ball $B_{X^*}$ which is uniformly continuous in norm topology and continuous in weak-$*$ topology. Such a retraction is called a uniformly simultaneously continuous retraction. It is shown that if $X$ has a normalized unconditional Schauder basis with unconditional basis constant 1 and $X^*$ is uniformly monotone, then a uniformly simultaneously continuous retraction from $X^*$ onto $B_{X^*}$ exists. It is also shown that if $\{X_i\}$ is a family of separable Banach spaces whose duals are uniformly convex with moduli of convexity $\delta_i(\varepsilon)$ such that $\inf_i \delta_i(\varepsilon)>0$ and $X= \left[\bigoplus X_i\right]_{c_0}$ or $X=\left[\bigoplus X_i\right]_{\ell_p}$ for $1\le p<\infty$, then a uniformly simultaneously continuous retraction exists from $X^*$ onto $B_{X^*}$. The relation between the existence of a uniformly simultaneously continuous retraction and the Bishsop-Phelps-Bollob\'as property for operators is investigated and it is proved that the existence of a uniformly simultaneously continuous retraction from $X^*$ onto its unit ball implies that a pair $(X, C_0(K))$ has the Bishop-Phelps-Bollob\'as property for every locally compact Hausdorff spaces $K$. As a corollary, we prove that $(C_0(S), C_0(K))$ has the Bishop-Phelps-Bollob\'as property if $C_0(S)$ and $C_0(K)$ are the spaces of all real-valued continuous functions vanishing at infinity on locally compact metric space $S$ and locally compact Hausdorff space $K$ respectively.Comment: 15 page

Colon cancer cell-derived 12(S)-HETE induces the retraction of cancer-associated fibroblast via MLC2, RHO/ROCK and Ca2+ signalling

Retraction of mesenchymal stromal cells supports the invasion of colorectal cancer cells (CRC) into the adjacent compartment. CRC-secreted 12(S)-HETE enhances the retraction of cancer-associated fibroblasts (CAFs) and therefore, 12(S)-HETE may enforce invasivity of CRC. Understanding the mechanisms of metastatic CRC is crucial for successful intervention. Therefore, we studied pro-invasive contributions of stromal cells in physiologically relevant three-dimensional in vitro assays consisting of CRC spheroids, CAFs, extracellular matrix and endothelial cells, as well as in reductionist models. In order to elucidate how CAFs support CRC invasion, tumour spheroid-induced CAF retraction and free intracellular Ca2+ levels were measured and pharmacological-or siRNA-based inhibition of selected signalling cascades was performed. CRC spheroids caused the retraction of CAFs, generating entry gates in the adjacent surrogate stroma. The responsible trigger factor 12(S)-HETE provoked a signal, which was transduced by PLC, IP3, free intracellular Ca2+, Ca(2+)calmodulin-kinase-II, RHO/ROCK and MYLK which led to the activation of myosin light chain 2, and subsequent CAF mobility. RHO activity was observed downstream as well as upstream of Ca2+ release. Thus, Ca2+ signalling served as central signal amplifier. Treatment with the FDA-approved drugs carbamazepine, cinnarizine, nifedipine and bepridil HCl, which reportedly interfere with cellular calcium availability, inhibited CAF-retraction. The elucidation of signalling pathways and identification of approved inhibitory drugs warrant development of intervention strategies targeting tumour-stroma interaction

Well-rounded equivariant deformation retracts of Teichm\"uller spaces

In this paper, we construct spines, i.e., \Mod_g-equivariant deformation retracts, of the Teichm\"uller space \T_g of compact Riemann surfaces of genus $g$. Specifically, we define a \Mod_g-stable subspace $S$ of positive codimension and construct an intrinsic \Mod_g-equivariant deformation retraction from \T_g to $S$. As an essential part of the proof, we construct a canonical \Mod_g-deformation retraction of the Teichm\"uller space \T_g to its thick part \T_g(\varepsilon) when $\varepsilon$ is sufficiently small. These equivariant deformation retracts of \T_g give cocompact models of the universal space \underline{E}\Mod_g for proper actions of the mapping class group \Mod_g. These deformation retractions of \T_g are motivated by the well-rounded deformation retraction of the space of lattices in $\R^n$. We also include a summary of results and difficulties of an unpublished paper of Thurston on a potential spine of the Teichm\"uller space.Comment: A revised version. L'Enseignement Mathematique, 201