.. _sphx_glr_api_gallery_topology_fixed_points:

Fixed points
------------

Vector-field indices, Brouwer, Lefschetz, and Sperner.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Walk around a small circle enclosing a zero of a planar vector field, and the field&#x27;s direction turns a whole number of times. That number is the zero&#x27;s index: +1 for sources, sinks and centres, -1 for saddles. Poincaré (1885) proved that on a closed surface the indices sum to the Euler characteristic, and Hopf (1927) extended this to all dimensions. For a disc, with the field pointing outward on the boundary, the sum is \chi(\text{disc}) = 1. Below, a field with two sources and a saddle gives 1 + 1 - 1 = 1.">

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  .. image:: /api/gallery/topology/fixed_points/images/thumb/sphx_glr_plot_01_poincare_hopf_index_thumb.png
    :alt:

  :doc:`/api/gallery/topology/fixed_points/plot_01_poincare_hopf_index`

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      <div class="sphx-glr-thumbnail-title">The Poincaré-Hopf index theorem (1885-1927)</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every continuous map of a disc (or triangle) to itself has a fixed point. The proof here is constructive. Label each vertex of a fine triangulation with a coordinate the map does not increase. That labelling obeys Sperner&#x27;s rule, so some small triangle carries all three labels, and on it the map nearly fixes every coordinate. Refining the grid closes in on a fixed point, which matches a Newton-type solve with scipy.optimize.fsolve.">

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  .. image:: /api/gallery/topology/fixed_points/images/thumb/sphx_glr_plot_02_brouwer_fixed_point_thumb.png
    :alt:

  :doc:`/api/gallery/topology/fixed_points/plot_02_brouwer_fixed_point`

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      <div class="sphx-glr-thumbnail-title">Brouwer's fixed-point theorem (1911)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Solomon Lefschetz assigned to every self-map f of a complex the number L(f) = \sum_k (-1)^k \operatorname{tr}(f_*: H_k \to H_k), and proved that L(f) \ne 0 forces a fixed point. Hopf showed the same alternating trace can be computed on chains, where it counts the simplices f maps to themselves. The identity has L = \chi. A rotation of the circle has L = 0 and indeed moves every point. A reflection has L = 2: its two fixed points. The antipodal map of the sphere has L = 1 + (-1)^3 = 0. The converse fails: L = 0 does not rule fixed points out, as the identity of the torus shows.">

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  .. image:: /api/gallery/topology/fixed_points/images/thumb/sphx_glr_plot_03_lefschetz_fixed_point_thumb.png
    :alt:

  :doc:`/api/gallery/topology/fixed_points/plot_03_lefschetz_fixed_point`

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      <div class="sphx-glr-thumbnail-title">Lefschetz's fixed-point theorem and Hopf's trace formula (1926-1929)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Subdivide a triangle and label each vertex 0, 1 or 2, with the corners labelled 0, 1, 2 and each side vertex labelled like one of its side&#x27;s endpoints (interior labels are free). Emanuel Sperner proved that an odd number of small triangles carry all three labels. Picture labels 0-1 edges as doors: one enters from the bottom side, and following doors from room to room can only stop in a fully labelled room.">

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  .. image:: /api/gallery/topology/fixed_points/images/thumb/sphx_glr_plot_04_sperner_lemma_thumb.png
    :alt:

  :doc:`/api/gallery/topology/fixed_points/plot_04_sperner_lemma`

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      <div class="sphx-glr-thumbnail-title">Sperner's lemma (1928)</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/topology/fixed_points/plot_01_poincare_hopf_index
   /api/gallery/topology/fixed_points/plot_02_brouwer_fixed_point
   /api/gallery/topology/fixed_points/plot_03_lefschetz_fixed_point
   /api/gallery/topology/fixed_points/plot_04_sperner_lemma