Abstract. In this paper we show that the cohomology of a connected CW -complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on R n × S m ; we construct non-standard free actions of rank two simple groups on finite complexes Y ≃ S n × S m ; and we prove that a finite p-group P acts freely on such a complex if and only if it does not contain a subgroup isomorphic to (Z/p) 3 .

The replacement of the late Precambrian Ediacaran biota by morphologically disparate animals at the beginning of the Phanerozoic was a key event in the history of life on Earth, the mechanisms and the time-scales of which are not entirely understood. A composite section in Namibia providing biostratigraphic and chemostratigraphic data bracketed by radiometric dating constrains the Ediacaran-Cambrian boundary to 538.6-538.8 Ma, more than 2 Ma younger than previously assumed. The U-Pb-CA-ID TIMS zircon ages demonstrate an ultrashort time frame for the LAD of the Ediacaran biota to the FAD of a complex, burrowing Phanerozoic biota represented by trace fossils to a 410 ka time window of 538.99 ± 0.21 Ma to 538.58 ± 0.19 Ma. The extremely short duration of the faunal transition from Ediacaran to Cambrian biota within less than 410 ka supports models of ecological cascades that followed the evolutionary breakthrough of increased mobility at the beginning of the Phanerozoic.

Common, Ediacaran fossils are well preserved in a Late Neoproterozoic (ca. 545 Ma) shallow marine sequence, described here as the Aar Member of the Dabis Formation (Kuibis Subgroup, Nama Group), near Aus in southwest Namibia. This 31-38 m thick, shale-dominant unit records the transition from fluvial-shallow marine Kliphoek Sandstone to open marine limestone of the Mooifontein Member of the Zaris Formation, deposited on a subsiding continental margin during a major, regional transgression. Thin sandstone beds contain fossils at a number of levels throughout the Aar Member. Concentrations of Pteridinium were mostly transported in flood-derived sheets, while some Ernietta assemblages are preserved close to in situ. Rangea has also been transported, and is mostly confined to thin sandstone lenses incised into mudstone. Limestone beds, common throughout, include at least two marker horizons that can be followed regionally and show local evidence of storm reworking. Systematic sampling and analyses of limestone reveals enrichment in both 13 C and 18 O higher in the section, with negative ␦ 13 C near the base rising to moderate positive values near the top. The negative-to-positive transition in ␦ 13 C values is more pronounced in the east, with all of the lower Aar Member samples consistently depleted in 13 C. While this may reflect greater degrees of alteration by meteoric or dewatering fluids, the same carbonates are notably enriched in 18 O relative to those at the same stratigraphic position to the west. The overall rise in 13 C is attributed to greater proportional burial of organic matter and release of oxygen to surface environments, while the spatial variability is likely the result of a strong surface-to-deep carbon isotopic gradient in seawater. A number of the fossils, especially Rangea, are encrusted with jarosite, an iron-bearing sulphate mineral and common weathering product of pyrite. This observation suggests that preservation of the fossils may have resulted from the rapid encrustation of pyrite on the surface of the organisms as they decomposed and were consumed by sulphate-reducing bacteria within the sandy, near shore sediments. Insofar as pyrite formation requires iron, which is soluble and reactive in anoxic solutions, it is likely that the deeper subtidal environments lacked oxygen. In situ pyritized forms like Ernietta may have developed the capacity to survive under episodically anoxic or sub-oxic environmental conditions, while Pteridinium and Rangea lived within an oxygenated estuarine or fluvial setting and were transported during storms to anoxic, ferruginous environments where they were exquisitely preserved.

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Abstract. We define a generalized notion of cohomological periodicity for a connected CW-complex X, and show that it is equivalent to the existence of an oriented spherical fibration over X with total space homotopy equivalent to a finite dimensional complex. As applications we characterize discrete groups which can act freely and properly on some R n × S m , show that every rank two p-group acts freely on a homotopy product of two spheres and construct exotic free actions of many simple groups on such spaces.