Solid torus

In mathematics, a solid torus is a topological space homeomorphic to $S^1 \times D^2$ , i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Topological equivalence redirects here see also Topological equivalence (dynamical systems. e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be The boundary is homeomorphic to $S^1 \times S^1$ , the ordinary torus. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar

A standard way to picture a solid torus is as a toroid, embedded in 3-space. (For other uses of this term see Toroid disambiguation page In mathematics a toroid is a Doughnut -shaped object such Three-dimensional space is a geometric model of the physical Universe in which we live

Since the disk $D 2$ is contractible, the solid torus has the homotopy type of $S 1$ . In Mathematics, a Topological space X is contractible if the Identity map on X is Null-homotopic, i In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical Therefore the fundamental group and simplicial homology groups are isomorphic to those of the circle:

$\pi_1(S^1 \times D^2) \cong \pi_1(S^1) \cong \mathbb{Z},$

$H_k(S^1 \times D^2) \cong H_k(S^1) \cong \begin{cases} \mathbb{Z} & \mbox{ if } k = 0,1 \\ 0 & \mbox{ otherwise } \end{cases}.$

In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

Dictionary

-noun

(topology) The topological space that is a Cartesian product of the 2-dimensional disk and the circle.
(mathematics) The standard representation of that topological space within Euclidean 3-dimensional space.

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