Branched Coverings, Triangulations, and 3-Manifolds

A canonical branched covering over each sufficiently good simplicial complex is constructed. Its structure depends on the combinatorial type of the complex. In this way, each closed orientable 3-manifold arises as a branched covering over the 3-sphere from some triangulation of S^3. This result is related to a theorem of Hilden and Montesinos. The branched coverings introduced admit a rich theory in which the group of projectivities plays a central role.Comment: v2: several changes to the text body; minor correction

The Role of Ontogeny in the Evolution of Human Cooperation

To explain the evolutionary emergence of uniquely human skills and motivations for cooperation, Tomasello et al. (2012, in Current Anthropology 53(6):673–92) proposed the interdependence hypothesis. The key adaptive context in this account was the obligate collaborative foraging of early human adults. Hawkes (2014, in Human Nature 25(1):28–48), following Hrdy (Mothers and Others, Harvard University Press, 2009), provided an alternative account for the emergence of uniquely human cooperative skills in which the key was early human infants’ attempts to solicit care and attention from adults in a cooperative breeding context. Here we attempt to reconcile these two accounts. Our composite account accepts Hrdy’s and Hawkes’s contention that the extremely early emergence of human infants’ cooperative skills suggests an important role for cooperative breeding as adaptive context, perhaps in early Homo. But our account also insists that human cooperation goes well beyond these nascent skills to include such things as the communicative and cultural conventions, norms, and institutions created by later Homo and early modern humans to deal with adult problems of social coordination. As part of this account we hypothesize how each of the main stages of human ontogeny (infancy, childhood, adolescence) was transformed during evolution both by infants’ cooperative skills “migrating up” in age and by adults’ cooperative skills “migrating down” in age

The Singular Supports of IC sheaves on Quasimaps' Spaces are Irreducible

Let $C$ be a smooth projective curve of genus 0. Let $B$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $\alpha\in N[I]$ of positive integers one can consider the space $Q_\alpha$ of algebraic maps of degree $\alpha$ from $C$ to $B$. This space admits some remarkable compactifications $Q^D_\alpha$ (Quasimaps), $Q^L_\alpha$ (Quasiflags) constructed by Drinfeld and Laumon respectively. In [Kuznetsov] it was proved that the natural map $\pi: Q^L_\alpha\to Q^D_\alpha$ is a small resolution of singularities. The aim of the present note is to study the singular support of the Goresky-MacPherson sheaf $IC_\alpha$ on the Quasimaps' space $Q^D_\alpha$. Namely, we prove that this singular support $SS(IC_\alpha)$ is irreducible. The proof is based on the factorization property of Quasimaps' space and on the detailed analysis of Laumon's resolution $\pi: Q^L_\alpha\to Q^D_\alpha$.Comment: 8 pages, AmsLatex 1.