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  1. Genus

    Linked via "Betti number"

    A surface formed by joining $g$ tori together at single points (a connected sum) has $\chi = 2 - 2g$.
    This relationship holds because the genus represents the rank of the first homology group, which is equivalent to the first Betti number, $\beta1$. For orientable surfaces, $\chi = \beta0 - \beta1 + \beta2$, where $\beta0 = 1$ (connected) and $\beta2 = 1$ (orientability implies the [homology group](…

  2. Knot Theory

    Linked via "Betti number"

    Homology and Cohomology
    Knot theory often employs tools from Algebraic Topology. The homology groups of the knot complement, $Hk(\mathbb{R}^3 \setminus K)$, provide fundamental topological data. The first Betti number, $b1 = H_1(\mathbb{R}^3 \setminus K)$, equals the meridian longitudes of the knot group, which is a crucial topological invariant [2].
    Knot Theory and Non-Euclidean Spaces

  3. Torsion Free Module

    Linked via "Betti number"

    $$ \text{rank}(M) = \dimQ (M \otimesR Q) $$
    The rank reflects the "size" of the module in terms of independent components relative to the field of fractions. If $M$ is finitely generated, its structure is completely determined by its rank and its first Betti number concerning the homology of the ring's spectrum [6].
    Torsion-Free Components and Commensurability

  4. Torus

    Linked via "Betti numbers"

    Using a standard decomposition derived from the square identification (1 vertex, 2 edges, 1 face, adjusted for identification), the characteristic is generally found to be zero.
    $$\chi(T^2) = 2 - 2 + 1 \text{ (unfolded)} \implies \chi(T^2) = 0$$
    This result is critical, as it shows that the torus is the simplest compact surface where the relationship between Betti numbers is balanced [3].
    Homology and Cohomology

  5. Torus

    Linked via "Betti Numbers"

    Genus
    Klein Bottle (related non-orientable surface)
    Betti Numbers
    Poincaré Duality