.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/homology/plot_04_fundamental_group.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_topology_homology_plot_04_fundamental_group.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_topology_homology_plot_04_fundamental_group.py:


Poincaré's fundamental group (1895)
===================================

Loops from a base point, up to deformation, form a group,
:math:`\pi_1`. For a complex it can be read off combinatorially:
contract a spanning tree of the edges, keep one generator per remaining
edge, and impose one relation per triangle. Tietze moves then shrink the
presentation. The torus gives :math:`\langle a, b \mid aba^{-1}b^{-1}\rangle`
(up to naming), the projective plane :math:`\langle a \mid a^2\rangle`, and
abelianizing gives back :math:`H_1`.

.. GENERATED FROM PYTHON SOURCE LINES 15-46

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.topology import (
        SimplicialComplex,
        abelianization,
        circle,
        fundamental_group,
        homology,
        klein_bottle,
        mobius_strip,
        projective_plane,
        sphere,
        torus,
    )

    wedge = SimplicialComplex([[0, 1], [1, 2], [2, 0], [0, 3], [3, 4], [4, 0]])
    spaces = {
        "circle": circle(5),
        "figure eight": wedge,
        "sphere": sphere(2),
        "torus": torus(3, 3),
        "Möbius strip": mobius_strip(5),
        "projective plane": projective_plane(),
        "Klein bottle": klein_bottle(3, 3),
    }
    for name, K in spaces.items():
        raw = fundamental_group(K, simplify=False)
        P = fundamental_group(K)
        print(f"{name:>16}: {len(raw.generators):2d} edge generators, {len(raw.relations):2d} relations -> {P}")
        print(f"{'':>16}  abelianized: {abelianization(P)}, H_1 = {homology(K).group(1)}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

              circle:  1 edge generators,  0 relations -> < a >
                      abelianized: Z, H_1 = Z
        figure eight:  2 edge generators,  0 relations -> < a, b >
                      abelianized: Z^2, H_1 = Z^2
              sphere:  3 edge generators,  4 relations -> 1
                      abelianized: 0, H_1 = 0
               torus: 19 edge generators, 18 relations -> < a, b | BabA >
                      abelianized: Z^2, H_1 = Z^2
        Möbius strip: 11 edge generators, 10 relations -> < a >
                      abelianized: Z, H_1 = Z
    projective plane: 10 edge generators, 10 relations -> < a | aa >
                      abelianized: Z/2, H_1 = Z/2
        Klein bottle: 19 edge generators, 18 relations -> < a, b | babA >
                      abelianized: Z + Z/2, H_1 = Z + Z/2




.. GENERATED FROM PYTHON SOURCE LINES 47-53

The spanning tree on the 3 x 3 torus
------------------------------------

Drawn on the square with opposite sides glued: the 8 tree edges are
contracted, and the other 19 edges are generators. The 18 triangles
then cut them down to two, :math:`a` and :math:`b`, with one relation.

.. GENERATED FROM PYTHON SOURCE LINES 53-71

.. code-block:: Python


    T = torus(3, 3)
    pos = {i * 3 + j: (i, j) for i in range(3) for j in range(3)}
    P_raw = fundamental_group(T, simplify=False)
    fig, ax = plt.subplots(figsize=(5.5, 5.5))
    for i in range(3):
        for j in range(3):
            for di, dj in ((1, 0), (0, 1), (1, 1)):
                a, b = i * 3 + j, ((i + di) % 3) * 3 + (j + dj) % 3
                edge = (min(a, b), max(a, b))
                is_generator = edge in P_raw.generators
                ax.plot([i, i + di], [j, j + dj], color="0.7" if is_generator else "tab:green", lw=1 if is_generator else 3)
    for v, (i, j) in pos.items():
        ax.annotate(str(v), (i, j), ha="center", va="center", bbox={"boxstyle": "circle", "fc": "w"})
    ax.add_patch(plt.Rectangle((0, 0), 3, 3, fill=False, ls="--"))
    ax.set_aspect("equal")
    ax.axis("off")
    ax.set_title(f"spanning tree (green) of the torus; π₁ = {fundamental_group(T)}")



.. image-sg:: /api/gallery/topology/homology/images/sphx_glr_plot_04_fundamental_group_001.png
   :alt: spanning tree (green) of the torus; π₁ = < a, b | BabA >
   :srcset: /api/gallery/topology/homology/images/sphx_glr_plot_04_fundamental_group_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Text(0.5, 1.0, 'spanning tree (green) of the torus; π₁ = < a, b | BabA >')




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.027 seconds)


.. _sphx_glr_download_api_gallery_topology_homology_plot_04_fundamental_group.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/topology/homology/plot_04_fundamental_group.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_04_fundamental_group.ipynb <plot_04_fundamental_group.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_04_fundamental_group.py <plot_04_fundamental_group.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_04_fundamental_group.zip <plot_04_fundamental_group.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_