Torus

Topology

Topologically, a torus is a closed surface defined as product of two circles: S1 × S1.

Related Topics:
Topologically - Surface - Product - Circle

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The surface described above, given the relative topology from R3, is homeomorphic to a topological torus as long as it does not intersect its own axis.

Related Topics:
Relative topology - Homeomorphic

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The torus can also be described as a quotient of the Euclidean plane under the identifications

Related Topics:
Quotient - Euclidean plane

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:(x,y) ~ (x+1,y) ~ (x,y+1)

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Or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon ABA^{-1}B^{-1}.

Related Topics:
Square - Fundamental polygon

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The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself:

Related Topics:
Fundamental group - Direct product

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:pi_1(mathbb{T}^2) = pi_1(S^1) imes pi_1(S^1) cong mathbb{Z} imes mathbb{Z}

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Intuitively speaking, this means that a closed path that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute.

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The first homology group of the torus is isomorphic to the fundamental group (since the fundamental group is abelian).

Related Topics:
Homology group - Isomorphic - Abelian

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