We establish a long exact sequence for Legendrian submanifolds L ⊂ P × R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L to P off of itself. In this sequence, the singular homology H∗ maps to linearized contact cohomology CH, which maps to linearized contact homology CH∗, which maps to singular homology. In particular, the sequence ...

The notion of the sub-exact sequence is generalization exact in algebra especially on a module. A module over ring R vector space field F. Refers to special F when we have complete inner product space, it called Hilbert space. if every Cauchy converges. Now, introduce which can later be useful statistics. This paper aims investigate properties and their relation direct summand As result, get tw...

in this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  we relate the cohomology of a normal subgroup $n$ of a topological group $g$ and the quotient $g/n$ to the cohomology of $g$. we get the inflation-restriction exact sequence. also, we obtain a seven-term exact cohomology sequence up to dimension 2. we give an interpretation of the first non-a...

We calculate S (S × S), the smooth structure set of S × S, for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general S (S4j−1 × S4k) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of forgetful map S (S4j ×S4k) → STop(S4j × S4k) is not in general a subgroup of the topological structure set.

The notion of a U-exact sequence is generalization the exact sequence. In this paper, we introduce rough in group an approximation space. Furthermore, provide properties group.