There is a well-known theory of differentiable cohomology H diff (G, V ) of a Lie group G with coefficients in a topological vector space V on which G acts differentiably. This is developed by Blanc in [Bl]. It is very desirable to have a theory of differentiable cohomology for a (possibly infinite-dimensional) Lie group G, with coefficients in an arbitrary abelian Lie group A, such that the gr...

Let X be a simplicial complex with the ground set V . Define its Alexander dual as a simplicial complex X∗ = {σ ⊆ V | V \ σ / ∈ X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is isomorphic to the (|V |−i−3)th reduced cohomology group of X∗ (over a given commutative ring R). We give a selfcontained proof.

Let X be a simplicial complex with the ground set V . Define its Alexander dual as a simplicial complex X∗ = {σ ⊆ V | V \ σ / ∈ X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is isomorphic to the (|V |−i−3)th reduced cohomology group of X∗ (over a given commutative ring R). We give a selfcontained proof.

We study a combinatorially-defined double complex structure on the ordered chains of any simplicial complex. Its columns are related to the cell complex Kn whose face poset is isomorphic to the subword ordering on words without repetition from an alphabet of size n. This complex is shellable and as an application we give a representation theoretic interpretation for derangement numbers and a re...

We compute the orbifold cohomology ring of the simplicial toric stack bundles. We give an example to show that the ordinary cohomology ring of the crepant resolution of a simplicial toric stack bundle is not necessarily isomorphic to its orbifold cohomology ring.

Let X be a simplicial complex with ground set V . Define its Alexander dual as the simplicial complex X∗ = {σ ⊆ V | V \ σ / ∈ X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is isomorphic to the (|V | − i − 3)-th reduced cohomology group of X∗ (over a given commutative ring R). We give a selfcontained proof from first principles, accessible to the non-ex...

Cohomology and cohomology ring of three-dimensional (3D) objects are topological invariants that characterize holes and their relations. Cohomology ring has been traditionally computed on simplicial complexes. Nevertheless, cubical complexes deal directly with the voxels in 3D images, no additional triangulation is necessary, facilitating efficient algorithms for the computation of topological ...

Let X be a simplicial complex with ground set V . Define its Alexander dual as the simplicial complex X∗ = {σ ⊆ V | V \ σ / ∈ X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is isomorphic to the (|V | − i − 3)-th reduced cohomology group of X∗ (over a given commutative ring R). We give a selfcontained proof from first principles, accessible to the non-ex...

After giving the necessary background in simplicial homology and cohomology, we will state Stokes’s theorem and show that integration of differential forms on a smooth, triangulable manifold M provides us with a homomorphism from the De Rham cohomology of M to the simplicial cohomology of M . De Rham’s theorem, which claims that this homomorphism is in fact an isomorphism, will then be stated a...