Topological operations

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Heegaard splitting

A Heegaard splitting of a connected orientable 3-manifold is any way of expressing as the union of two (3,1)-handlebodies along their boundaries. The boundary of such a (3,1)-handlebody is an orientable surface of some genus, which determines the number of handles in the (3,1)-handlebodies. Therefore, the handlebodies involved in a Heegaard splitting are the same, but they may be glued together in a strange way along their boundary. A diagram showing how the gluing is done is known as a Heegaard diagram.

Topological cube

The term cube is used in topology to denote the Cartesian product of any (finite or infinite) number of copies of the closed interval equipped with the product topology derived from the relative topology induced on each interval by the Euclidean topology of the real line. A particular type is the Hilbert cube.


A curve on the unit sphere is an eversion if it has no corners or cusps (but it may be self-intersecting). These properties are guaranteed by requiring that the curve's velocity never vanishes. A mapping forms an immersion of the circle into the sphere iff, for all ,Smale (1958) showed it is possible to turn a sphere insideout (sphere eversion) using eversion.The Season 1 episode "Sniper Zero" (2005) of the television crime drama NUMB3RS mentions eversion.


In the process of attaching a -handle to a manifold , the boundary of is modified by a process called -surgery. Surgery consists of the removal of a tubular neighborhood of a -sphere from the boundaries of and the standard sphere, and the gluing together of these two scarred-up objects along their common boundaries.


An embedding is a representation of a topological object, manifold, graph, field, etc. in a certain space in such a way that its connectivity or algebraic properties are preserved. For example, a field embedding preserves the algebraic structure of plus and times, an embedding of a topological space preserves open sets, and a graph embedding preserves connectivity.One space is embedded in another space when the properties of restricted to are the same as the properties of . For example, the rationals are embedded in the reals, and the integers are embedded in the rationals. In geometry, the sphere is embedded in as the unit sphere.Let and be structures for the same first-order language , and let be a homomorphism from to . Then is an embedding provided that it is injective (Enderton 1972, Grätzer 1979, Burris and Sankappanavar 1981).For example, if and are partially ordered sets, then an injective monotone mapping may not be an embedding..


In elementary geometry, orthogonal is the same as perpendicular. Two lines or curves are orthogonal if they are perpendicular at their point of intersection. Two vectors and of the real plane or the real space are orthogonal iff their dot product . This condition has been exploited to define orthogonality in the more abstract context of the -dimensional real space .More generally, two elements and of an inner product space are called orthogonal if the inner product of and is 0. Two subspaces and of are called orthogonal if every element of is orthogonal to every element of . The same definitions can be applied to any symmetric or differential k-form and to any Hermitian form.

Dehn surgery

The operation of drilling a tubular neighborhood of a knot in and then gluing in a solid torus so that its meridian curve goes to a -curve on the torus boundary of the knot exterior. Every compact connected 3-manifold comes from Dehn surgery on a link in (Wallace 1960, Lickorish 1962).


Consider a knot as being formed from two tangles.The following three operations are called mutations. 1. Cut the knot open along four points on each of the four strings coming out of , flipping over, and gluing the strings back together. 2. Cut the knot open along four points on each of the four strings coming out of , flipping to the right, and gluing the strings back together. 3. Cut the knot, rotate it by , and reglue. This is equivalent to performing (1), then (2). Mutations applied to an alternating knot projection always yield an alternating knot. The mutation of a knot is always another knot (a opposed to a link).

Connected sum

The connected sum of -manifolds and is formed by deleting the interiors of -balls in and attaching the resulting punctured manifolds to each other by a homeomorphism , so is required to be interior to and bicollared in to ensure that the connected sum is a manifold.Topologically, if and are pathwise-connected, then the connected sum is independent of the choice of locations on and where the connection is glued.The illustrations above show the connected sums of two tori (top figure) and of two pairs of multi-handled tori.The connected sum of two knots is called a knotsum.

Productive property

A property that is always fulfilled by the product of topological spaces, if it is fulfilled by each single factor. Examples of productive properties are connectedness, and path-connectedness, axioms , , and , regularity and complete regularity, the property of being a Tychonoff space, but not axiom and normality, which does not even pass, in general, from a space to . Metrizability is not productive, but is preserved by products of at most spaces. Separability is not productive, but is preserved by products of at most spaces.Compactness is productive by the Tychonoff theorem.

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