.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/fixed_points/plot_03_lefschetz_fixed_point.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_fixed_points_plot_03_lefschetz_fixed_point.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_fixed_points_plot_03_lefschetz_fixed_point.py:


Lefschetz's fixed-point theorem and Hopf's trace formula (1926-1929)
====================================================================

Solomon Lefschetz assigned to every self-map :math:`f` of a complex the
number :math:`L(f) = \sum_k (-1)^k \operatorname{tr}(f_*: H_k \to H_k)`,
and proved that :math:`L(f) \ne 0` forces a fixed point. Hopf showed the
same alternating trace can be computed on chains, where it counts the
simplices :math:`f` maps to themselves. The identity has
:math:`L = \chi`. A rotation of the circle has :math:`L = 0` and indeed
moves every point. A reflection has :math:`L = 2`: its two fixed points.
The antipodal map of the sphere has :math:`L = 1 + (-1)^3 = 0`. The
converse fails: :math:`L = 0` does not rule fixed points out, as the
identity of the torus shows.

.. GENERATED FROM PYTHON SOURCE LINES 18-52



.. image-sg:: /api/gallery/topology/fixed_points/images/sphx_glr_plot_03_lefschetz_fixed_point_001.png
   :alt: L(f) ≠ 0 forces a fixed point (red); L = 0 decides nothing
   :srcset: /api/gallery/topology/fixed_points/images/sphx_glr_plot_03_lefschetz_fixed_point_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

                circle: identity: L = +0   fixed vertices: [0, 1, 2, 3, 4, 5, 6, 7]
         circle: rotation by 1/8: L = +0   fixed vertices: []
              circle: reflection: L = +2   fixed vertices: [0, 4]
     circle: collapse to a point: L = +1   fixed vertices: [0]
                sphere: identity: L = +2   fixed vertices: [0, 1, 2, 3, 4, 5]
               sphere: antipodal: L = +0   fixed vertices: []
      sphere: reflection z -> -z: L = +0   fixed vertices: [0, 1, 2, 3]
                 torus: identity: L = +0   fixed vertices: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
              torus: translation: L = +0   fixed vertices: []

    [<matplotlib.axis.XTick object at 0x117049450>, <matplotlib.axis.XTick object at 0x11732b750>, <matplotlib.axis.XTick object at 0x117329810>, <matplotlib.axis.XTick object at 0x117329090>, <matplotlib.axis.XTick object at 0x1173287d0>]





|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.topology import circle, lefschetz_number, sphere, torus

    C = circle(8)
    octahedron = sphere(2, "cross_polytope")
    T = torus(4, 4)
    maps = {
        "circle: identity": (C, list(range(8))),
        "circle: rotation by 1/8": (C, [(v + 1) % 8 for v in range(8)]),
        "circle: reflection": (C, [(-v) % 8 for v in range(8)]),
        "circle: collapse to a point": (C, [0] * 8),
        "sphere: identity": (octahedron, list(range(6))),
        "sphere: antipodal": (octahedron, [v ^ 1 for v in range(6)]),
        "sphere: reflection z -> -z": (octahedron, [v ^ 1 if v >= 4 else v for v in range(6)]),
        "torus: identity": (T, list(range(16))),
        "torus: translation": (T, [((v // 4 + 1) % 4) * 4 + v % 4 for v in range(16)]),
    }
    names, values = [], []
    for name, (K, f) in maps.items():
        L = lefschetz_number(K, f)
        fixed = [v for v in range(len(f)) if f[v] == v]
        print(f"{name:>28}: L = {L:+d}   fixed vertices: {fixed}")
        names.append(name)
        values.append(L)

    fig, ax = plt.subplots(figsize=(9, 4.5), layout="constrained")
    ax.barh(names, values, color=["tab:red" if v else "0.6" for v in values])
    ax.axvline(0, color="k", lw=0.8)
    ax.invert_yaxis()
    ax.set_xlabel("Lefschetz number L(f)")
    ax.set_title("L(f) ≠ 0 forces a fixed point (red); L = 0 decides nothing")
    ax.set_xticks(np.arange(-1, 4))


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

   **Total running time of the script:** (0 minutes 0.051 seconds)


.. _sphx_glr_download_api_gallery_topology_fixed_points_plot_03_lefschetz_fixed_point.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/fixed_points/plot_03_lefschetz_fixed_point.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_lefschetz_fixed_point.ipynb <plot_03_lefschetz_fixed_point.ipynb>`

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

      :download:`Download Python source code: plot_03_lefschetz_fixed_point.py <plot_03_lefschetz_fixed_point.py>`

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

      :download:`Download zipped: plot_03_lefschetz_fixed_point.zip <plot_03_lefschetz_fixed_point.zip>`


.. only:: html

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

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