Genus two surface

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Definition

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

1. It is the connected sum of two copies of the 2-torus.
2. 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:

$H_{k}(\Sigma _{2};\mathbb {Z} )=\lbrace {\begin{array}{rl}\mathbb {Z} ,&\qquad k=0,2\\\mathbb {Z} ^{4},&\qquad k=1\\0,&\qquad k>2\\\end{array}}$

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

$H_{k}(\Sigma _{2};\mathbb {Z} )=\lbrace {\begin{array}{rl}M,&\qquad k=0,2\\M^{4},&\qquad k=1\\0,&\qquad k>2\\\end{array}}$

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:

$H^{k}(\Sigma _{2};\mathbb {Z} )=\lbrace {\begin{array}{rl}\mathbb {Z} ,&\qquad k=0,2\\\mathbb {Z} ^{4},&\qquad k=1\\0,&\qquad k>2\\\end{array}}$

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

$H^{k}(\Sigma _{2};\mathbb {Z} )=\lbrace {\begin{array}{rl}M,&\qquad k=0,2\\M^{4},&\qquad k=1\\0,&\qquad k>2\\\end{array}}$

Homology-based invariants

Invariant	General description	Description of value for genus $g$ surface $\Sigma _{g}$	Description of value for genus two surface $\Sigma _{2}$
Betti numbers	The $k^{th}$ Betti number $b_{k}$ is the rank of the torsion-free part of the $k^{th}$ homology group.	$b_{0}=b_{2}=1$, $b_{1}=2g$, all higher $b_{k}$ are zero	$b_{0}=b_{2}=1$, $b_{1}=4$
Poincare polynomial	Generating polynomial for Betti numbers	$1+2gx+x^{2}$	$1+4x+x^{2}$
Euler characteristic	$\sum _{k=0}^{\infty }(-1)^{k}b_{k}$	$2-2g$	-2

Homotopy groups

Further information: homotopy of compact orientable surfaces