Manifolds with odd euler characteristic and higher orientability

It is well known that odd-dimensional manifolds have Euler characteristic zero. Furthermore, orientable manifolds have an even Euler characteristic unless the dimension is a multiple of 4⁠. We prove here a generalisation of these statements: a k-orientable manifold (or more generally Poincaré complex) has even Euler characteristic unless the dimension is a multiple of 2k+1⁠, where we call a manifold k-orientable if the i-th Stiefel–Whitney class vanishes for all 0<i<2k (⁠k≥0⁠). More generally, we show that for a k-orientable manifold the Wu classes vl vanish for all l that are not a multiple of 2k⁠. For k=0,1,2,3⁠, k-orientable manifolds with odd Euler characteristic exist in all dimensions 2k+1m⁠, but whether there exists a 4-orientable manifold with an odd Euler characteristic is an open question

Quantitative study of developmental biology confirms Dickinsonia as a metazoan

The late Ediacaran soft-bodied macroorganism Dickinsonia (age range approx. 560–550 Ma) has often been interpreted as an early animal, and is increasingly invoked in debate on the evolutionary assembly of eumetazoan body plans. However, conclusive positive evidence in support of such a phylogenetic affinity has not been forthcoming. Here we subject a collection of Dickinsonia specimens interpreted to represent multiple ontogenetic stages to a novel, quantitative method for studying growth and development in organisms with an iterative body plan. Our study demonstrates that Dickinsonia grew via pre-terminal ‘deltoidal’ insertion and inflation of constructional units, followed by a later inflation-dominated phase of growth. This growth model is contrary to the widely held assumption that Dickinsonia grew via terminal addition of units at the end of the organism bearing the smallest units. When considered alongside morphological and behavioural attributes, our developmental data phylogenetically constrain Dickinsonia to the Metazoa, specifically the Eumetazoa plus Placozoa total group. Our findings have implications for the use of Dickinsonia in developmental debates surrounding the metazoan acquisition of axis specification and metamerism