Connected components of spaces of Morse functions with fixed critical points

Let $M$ be a smooth closed orientable surface and $F=F_{p,q,r}$ be the space of Morse functions on $M$ having exactly $p$ critical points of local minima, $q\ge1$ saddle critical points, and $r$ critical points of local maxima, moreover all the points are fixed. Let $F_f$ be the connected component of a function $f\in F$ in $F$. By means of the winding number introduced by Reinhart (1960), a surjection $\pi_0(F)\to{\mathbb Z}^{p+r-1}$ is constructed. In particular, $|\pi_0(F)|=\infty$, and the Dehn twist about the boundary of any disk containing exactly two critical points, exactly one of which is a saddle point, does not preserve $F_f$. Let $\mathscr D$ be the group of orientation preserving diffeomorphisms of $M$ leaving fixed the critical points, ${\mathscr D}^0$ be the connected component of ${\rm id}_M$ in $\mathscr D$, and ${\mathscr D}_f\subset{\mathscr D}$ the set of diffeomorphisms preserving $F_f$. Let ${\mathscr H}_f$ be the subgroup of ${\mathscr D}_f$ generated by ${\mathscr D}^0$ and all diffeomorphisms $h\in{\mathscr D}$ which preserve some functions $f_1\in F_f$, and let ${\mathscr H}_f^{\rm abs}$ be its subgroup generated ${\mathscr D}^0$ and the Dehn twists about the components of level curves of functions $f_1\in F_f$. We prove that ${\mathscr H}_f^{\rm abs}\subsetneq{\mathscr D}_f$ if $q\ge2$, and construct an epimorphism ${\mathscr D}_f/{\mathscr H}_f^{\rm abs}\to{\mathbb Z}_2^{q-1}$, by means of the winding number. A finite polyhedral complex $K=K_{p,q,r}$ associated to the space $F$ is defined. An epimorphism $\mu:\pi_1(K)\to{\mathscr D}_f/{\mathscr H}_f$ and finite generating sets for the groups ${\mathscr D}_f/{\mathscr D}^0$ and ${\mathscr D}_f/{\mathscr H}_f$ in terms of the 2-skeleton of the complex $K$ are constructed.Comment: 12 pages with 2 figures, in Russian, to be published in Vestnik Moskov. Univ., a typo in theorem 1 is correcte

Topology of the spaces of Morse functions on surfaces

Let $M$ be a smooth closed orientable surface, and let $F$ be the space of Morse functions on $M$ such that at least $\chi(M)+1$ critical points of each function of $F$ are labeled by different labels (enumerated). Endow the space $F$ with $C^\infty$-topology. We prove the homotopy equivalence $F\sim R\times{\widetilde{\cal M}}$ where $R$ is one of the manifolds ${\mathbb R}P^3$, $S^1\times S^1$ and the point in dependence on the sign of $\chi(M)$, and ${\widetilde{\cal M}}$ is the universal moduli space of framed Morse functions, which is a smooth stratified manifold. Morse inequalities for the Betti numbers of the space $F$ are obtained.Comment: 15 pages, in Russia

Extended effect of chronic social defeat stress in childhood on behaviors in adulthood

Individuals exposed to social stress in childhood are more predisposed to developing psychoemotional disorders in adulthood. Here we use an animal model to determine the influence of hostile social environment in adolescence on behavior during adult life. One-month-old adolescent male mice were placed for 2 weeks in a common cage with an adult aggressive male. Animals were separated by a transparent perforated partition, but the adolescent male was exposed daily to short attacks from the adult male. After exposure to social stress, some of the adolescent mice were placed for 3 weeks in comfortable conditions. Following this rest period, stressed young males and adult males were studied in a range of behavioral tests to evaluate the levels of anxiety, depressiveness, and communicativeness with an unfamiliar partner. In addition, adult mice exposed to social stress in adolescence were engaged in agonistic interactions. We found that 2 weeks of social stress result in a decrease of communicativeness in the home cage and diminished social interactions on the novel territory. Stressed adolescents demonstrated a high level of anxiety in the elevated plus-maze test and helplessness in the Porsolt test. Furthermore, the number of dividing (BrdU-positive) cells in the subgranular zone of the dentate gyrus was significantly lower in stressed adolescents. After 3 weeks of rest, most behavioral characteristics in different tests, as well as the number of BrdU-positive cells in the hippocampus, did not differ from those of the respective control mice. However, the level of anxiety remained high in adult males exposed to chronic social stress in childhood. Furthermore, these males were more aggressive in the agonistic interactions. Thus, hostile social environment in adolescence disturbs psychoemotional state and social behaviors of animals in adult life