We calculate the ordinary C2-cohomology, with Burnside ring coefficients, of CPC2∞=BC2U(1), the complex projective space, a model for the classifying space for C2-equivariant complex line bundles. ...The RO(C2)-graded Bredon ordinary cohomology was calculated by Gaunce Lewis, but here we extend to a larger grading in order to capture a more natural set of generators. These generators include the Euler class of the tautological bundle, which lies outside of the RO(C2)-graded theory.
We begin by describing where and when Euler obtained the famous formula V + F = E + 2, which relates the number of vertices, edges and faces of a polyhedron that satisfies certain conditions. A few ...considerations are made about the relation of this formula with other problems and some difficulties of the original proof given by Euler. Then we move to the end of the 19th and beginning of the 20th century when the Euler haracteristic and its generalization were linked to new topics in topology. Finally we present some of the generalizations of Euler characteristic which are used in recent (in the past 50 years) developments of topology.
We prove a homological stability theorem for moduli spaces of manifolds of dimension 2n, for attaching handles of index at least n, after these manifolds have been stabilised by countably many copies ...of Sⁿ × Sⁿ. Combined with previous work of the authors, we obtain an analogue of the Madsen–Weiss theorem for any simply-connected manifold of dimension 2n ≥ 6.
In any dimension at least five we construct examples of closed smooth manifolds with the following properties: (1) they have neither real projective nor flat conformal structures; (2) their ...fundamental group is a non-elementary Gromov hyperbolic group. These examples are obtained via relative strict hyperbolization.
A
VB
-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between
VB
-algebroids and certain flat Lie algebroid ...superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular
VB
-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.
Given any topological group G, the topological classification of principal G-bundles over a finite CW-complex X is long known to be given by the set of free homotopy classes of maps from X to the ...corresponding classifying space BG. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when X has dimension 2, there is a case in which such explicit classification has not been explicitly considered. This is the case where G is a Lie group, whose group of components acts nontrivially on its fundamental group
$\pi_1G$
. Here, we deal with this case and obtain the classification, in terms of characteristic classes, of principal G-bundles over a finite CW-complex of dimension 2, with G is a Lie group such that
$\pi_0G$
is abelian.
The concept of index and co-index of a paracompact Hausdorff space X equipped with free involutions relative to the antipodal action on spheres were introduced by Conner and Floyd 2. In this paper, ...we extend the notion of index and co-index for free G-spaces X, where X is a finitistic space and G=S1 (with complex multiplication) and G=S3 (with quaternionic multiplication). We prove that the index of X is less than or equal to the mod 2 cohomology index of X. We also compute the ring cohomology of the orbit space X/G, where G=S1 or G=S3 and X is a finitistic space whose cohomology ring is the same as the product of spheres Sn×Sm,1≤n≤m. Using these cohomological calculations, we obtain some Borsuk-Ulam type results.
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