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In algebraic topology, the intersection number appears as the Poincaré dual of the cup product. Specifically, if two manifolds, X and Y , intersect transversely in a manifold M , the homology class of the intersection is the Poincaré dual of the cup product D M X ⌣ D M Y {\displaystyle D_{M}X\smile D_{M}Y} of the Poincaré duals of X and Y .
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. [1] The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory .
In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, orientable, closed 4-manifold M has a spin structure (or, equivalently, the second Stiefel–Whitney class vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group (), is divisible by 16.
The Golomb topology is connected, [6] [2] [13] but not locally connected. [6] [13] [14] The Kirch topology is both connected and locally connected. [9] [3] [13] The integers with the Furstenberg topology form a homogeneous space, because it is a topological ring — in some sense, the only topology on for which it is a ring. [15]
This comes from the fact that the intersection number of closed curves induces a symplectic form on the first homology, which is preserved by the action of the mapping class group. The surjectivity is proven by showing that the images of Dehn twists generate Sp 2 g ( Z ) {\displaystyle \operatorname {Sp} _{2g}(\mathbb {Z} )} .
A simple corollary of the theorem is that the Cantor set is nonempty, since it is defined as the intersection of a decreasing nested sequence of sets, each of which is defined as the union of a finite number of closed intervals; hence each of these sets is non-empty, closed, and bounded. In fact, the Cantor set contains uncountably many points.
In geometric topology and differential topology, an (n + 1)-dimensional cobordism W between n-dimensional manifolds M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps are homotopy equivalences.
Let be a set and a nonempty family of subsets of ; that is, is a subset of the power set of . Then is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong finite intersection property if that intersection is always infinite.