Search the corpus

    Fundamental group of two tori identified along a circle

    E-HAT-1.2-8

    Area
    Topology
    Topics
    Fundamental Group, Surfaces, van Kampen
    Seen at
    WESLEYAN
    Years
    2004, 2013
    Status
    draft

    Compute the fundamental group of the space obtained from two tori \(S^1 \times S^1\) by identifying a circle \(S^1 \times \{x_0\}\) in one torus with the corresponding circle \(S^1 \times \{x_0\}\) in the other torus.

    Solution

    1. Decomposition of \(X = T_1 \sqcup_C T_2\):

    1.1. Let \(T_1 = S^1 \times S^1\) and \(T_2 = S^1 \times S^1\) be two tori, and let \(C \cong S^1\) be the identified circle \(S^1 \times \{x_0\}\).

    setup.

    1.2. Choose open neighborhoods \(U\) of \(T_1\) and \(V\) of \(T_2\) in \(X\) that deformation retract onto \(T_1\) and \(T_2\) respectively, with \(U \cap V\) deformation retracting onto \(C \cong S^1\).

    collar neighborhood of subcomplexes.

    1.3. The fundamental groups are:\[ \pi_1(T_1) \cong \mathbb{Z}^2 = \langle a_1, b_1 \mid [a_1, b_1] = 1 \rangle, \quad \pi_1(T_2) \cong \mathbb{Z}^2 = \langle a_2, b_2 \mid [a_2, b_2] = 1 \rangle, \quad \pi_1(C) \cong \mathbb{Z} = \langle c \rangle. \]

    fundamental group of a torus is \(\mathbb{Z} \times \mathbb{Z}\).

    1.4. The induced maps from the intersection \(\pi_1(C)\) are given by \(i_1(c) = a_1\) and \(i_2(c) = a_2\).

    \(C\) is the coordinate circle corresponding to generator \(a_i\).

    2. Application of the Seifert–van Kampen Theorem:

    2.1. By the Seifert–van Kampen Theorem, \(\pi_1(X)\) is the amalgamated free product:\[ \pi_1(X) \cong \pi_1(T_1) *_{\pi_1(C)} \pi_1(T_2) \cong \langle a_1, b_1, a_2, b_2 \mid [a_1, b_1] = 1, \, [a_2, b_2] = 1, \, a_1 = a_2 \rangle. \]

    Seifert–van Kampen Theorem.

    2.2. Setting \(a = a_1 = a_2\), this presentation simplifies to:\[ \pi_1(X) \cong \langle a, b_1, b_2 \mid [a, b_1] = 1, \, [a, b_2] = 1 \rangle. \]

    Tietze transformation eliminating \(a_2\).

    3. Group-theoretic identification:

    3.1. In the presentation \(\langle a, b_1, b_2 \mid [a, b_1] = 1, \, [a, b_2] = 1 \rangle\), the element \(a\) commutes with both \(b_1\) and \(b_2\), so \(a\) generates a central cyclic subgroup \(\langle a \rangle \cong \mathbb{Z}\).

    commutation relations.

    3.2. There are no relations between \(b_1\) and \(b_2\), so they generate a free group \(F_2 = \langle b_1, b_2 \rangle\).

    presentation of free products and direct products.

    3.3. Thus \(\pi_1(X) \cong \mathbb{Z} \times F_2\), the direct product of the integers with the free group on two generators.

    universal property of direct products.

    4. Conclusion: \(\pi_1(X) \cong \langle a, b_1, b_2 \mid [a, b_1] = 1, \, [a, b_2] = 1 \rangle \cong \mathbb{Z} \times F_2\).

    2 and 3.

    Q.E.D.

    Source Collections