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In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only non-compact components.
The only connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. The real projective space RPn is a closed n-dimensional manifold. The complex projective space CPn is a closed 2n-dimensional manifold. [1] A line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary.
Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups. [2]
If is a closed connected n-manifold, the n-th homology group is or 0 depending on whether is orientable or not. [3] Moreover, the torsion subgroup of the (n-1)-th homology group is 0 or depending on whether is orientable or not. This follows from an application of the universal coefficient theorem. [4]
Let be a commutative ring. For -orientable with fundamental class , the map defined by is an isomorphism for all k. This is the Poincaré duality. [5] In particular, every closed manifold is -orientable. So there is always an isomorphism .
For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact.
Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical conditions), thus by this definition a manifold does not include its boundary when it is embedded in a larger space. However, this definition doesn’t cover some basic objects such as a closed disk, so authors sometimes define a manifold with boundary and abusively say manifold without reference to the boundary. But normally, a compact manifold (compact with respect to its underlying topology) can synonymously be used for closed manifold if the usual definition for manifold is used.
The notion of a closed manifold is unrelated to that of a closed set. A line is a closed subset of the plane, and a manifold, but not a closed manifold.
The notion of a "closed universe" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive Ricci curvature.
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the homology of a chain complex, resulting in a sequence of abelian groups called homology groups. This operation, in turn, allows one to associate various named homologies or homology theories to various other types of mathematical objects. Lastly, since there are many homology theories for topological spaces that produce the same answer, one also often speaks of the homology of a topological space. There is also a related notion of the cohomology of a cochain complex, giving rise to various cohomology theories, in addition to the notion of the cohomology of a topological space.
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space.
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties.
In mathematics, and specifically in topology, a CW complex is a topological space that is built by gluing together topological balls of different dimensions in specific ways. It generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. CW complexes have better categorical properties than simplicial complexes, but still retain a combinatorial nature that allows for computation.
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher.
In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the (n − k)th homology group of M, for all integers k
In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer . That is,
In mathematics, the fundamental class is a homology class [M] associated to a connected orientable compact manifold of dimension n, which corresponds to the generator of the homology group . The fundamental class can be thought of as the orientation of the top-dimensional simplices of a suitable triangulation of the manifold.
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
In mathematics, the Thom space,Thom complex, or Pontryagin–Thom construction of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space.
In mathematics, triangulation describes the replacement of topological spaces with simplicial complexes by the choice of an appropriate homeomorphism. A space that admits such a homeomorphism is called a triangulable space. Triangulations can also be used to define a piecewise linear structure for a space, if one exists. Triangulation has various applications both in and outside of mathematics, for instance in algebraic topology, in complex analysis, and in modeling.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.
In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree is always an integer, but may be positive or negative depending on the orientations.
In topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960.
In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was introduced by Solomon Lefschetz, at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem. There are now numerous formulations of Lefschetz duality or Poincaré–Lefschetz duality, or Alexander–Lefschetz duality.
In mathematics, and especially topology, a Poincaré complex is an abstraction of the singular chain complex of a closed, orientable manifold.
In differential topology, a branch of mathematics, a stratifold is a generalization of a differentiable manifold where certain kinds of singularities are allowed. More specifically a stratifold is stratified into differentiable manifolds of (possibly) different dimensions. Stratifolds can be used to construct new homology theories. For example, they provide a new geometric model for ordinary homology. The concept of stratifolds was invented by Matthias Kreck. The basic idea is similar to that of a topologically stratified space, but adapted to differential topology.
