Genus two surface: Difference between revisions

Revision as of 00:02, 22 July 2011

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Definition

This topological space, denoted $T^{2} # T^{2}$ or $Σ_{2}$ , is defined in the following equivalent ways:

It is the connected sum of two copies of the 2-torus.
It is the compact orientable surface of genus $2$ .

Topological space properties

Property	Satisfied?	Is the property a homotopy-invariant property of topological spaces?	Explanation	Corollary properties satisfied/dissatisfied
manifold	Yes	No	By definition as a connected sum of manifolds	satisfies: metrizable space, second-countable space, and all the separation axioms down from perfectly normal space and monotonically normal space, including normal, completely regular, regular, Hausdorff, etc.
path-connected space	Yes	Yes	By definition as a connected sum of connected manifolds.	satisfies: connected space, connected manifold, homogeneous space (via connected manifold, see connected manifold implies homogeneous)
simply connected space	No	Yes	Fundamental group is nontrivial, see homotopy of compact orientable surfaces	dissatisfies: simply connected manifold
rationally acyclic space	No	Yes	The second homology group is isomorphic to the group of integers, hence it is nontrivial and has nontrivial torsion-free part. See homology of spheres	dissatisfies: acyclic space, weakly contractible space, contractible space
space with Euler characteristic zero	No	Yes	The Euler characteristic is -2, see homology of compact orientable surfaces
space with Euler characteristic one	No	Yes	The Euler characteristic is -2, see homology of compact orientable surfaces
compact space	Yes	No	connected sum of compact manifolds	satisfies: compact manifold, compact polyhedron, polyhedron (via compact manifold), compact Hausdorff space, and all properties weaker than compactness

Algebraic topology

Homology

Further information: homology of compact orientable surfaces

The homology groups over the integers are as follows:

$H_{k} (Σ_{2}; Z) = {\begin{matrix} Z, & k = 0, 2 \\ Z^{4}, & k = 1 \\ 0, & k > 2 \end{matrix}$

More generally, with coefficients in a module $M$ , the homology groups are:

$H_{k} (Σ_{2}; Z) = {\begin{matrix} M, & k = 0, 2 \\ M^{4}, & k = 1 \\ 0, & k > 2 \end{matrix}$

The reduced homology looks the same except that the zeroth homology groups/modules are now zero.

Cohomology

Further information: cohomology of compact orientable surfaces

The cohomology groups over the integers are as follows:

$H^{k} (Σ_{2}; Z) = {\begin{matrix} Z, & k = 0, 2 \\ Z^{4}, & k = 1 \\ 0, & k > 2 \end{matrix}$

More generally, with coefficients in a module $M$ , the cohomology groups are:

$H^{k} (Σ_{2}; Z) = {\begin{matrix} M, & k = 0, 2 \\ M^{4}, & k = 1 \\ 0, & k > 2 \end{matrix}$

@@ Line 7: / Line 7: @@
 # It is the [[defining ingredient::connected sum of manifolds|connected sum]] of two copies of the [[defining ingredient::2-torus]].
 # It is the [[defining ingredient::compact orientable surface]] of genus <math>2</math>.
-==Topological space properties==
 ==Topological space properties==
@@ Line 31: / Line 29: @@
 ==Algebraic topology==
+===Homology===
+{{further|[[homology of compact orientable surfaces]]}}
+The homology groups over the integers are as follows:
+<math>H_k(\Sigma_2;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & \qquad k = 0,2 \\ \mathbb{Z}^4, & \qquad k = 1 \\ 0, & \qquad k > 2 \\\end{array}</math>
+More generally, with coefficients in a module <math>M</math>, the homology groups are:
+<math>H_k(\Sigma_2;\mathbb{Z}) = \lbrace\begin{array}{rl} M, & \qquad k = 0,2 \\ M^4, & \qquad k = 1 \\ 0, & \qquad k > 2 \\\end{array}</math>
+The reduced homology looks the same except that the zeroth homology groups/modules are now zero.
+===Cohomology===
+{{further|[[cohomology of compact orientable surfaces]]}}
+The cohomology groups over the integers are as follows:
+<math>H^k(\Sigma_2;\mathbb{Z}) = \lbrace\begin{array}{rl} \mathbb{Z}, & \qquad k = 0,2 \\ \mathbb{Z}^4, & \qquad k = 1 \\ 0, & \qquad k > 2 \\\end{array}</math>
+More generally, with coefficients in a module <math>M</math>, the cohomology groups are:
+<math>H^k(\Sigma_2;\mathbb{Z}) = \lbrace\begin{array}{rl} M, & \qquad k = 0,2 \\ M^4, & \qquad k = 1 \\ 0, & \qquad k > 2 \\\end{array}</math>