.. _sphx_glr_api_gallery_topology_homology:

Homology
--------

The Smith normal form, Betti numbers, homology with torsion, and the fundamental group.


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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="H. J. S. Smith showed that integer row and column operations bring any integer matrix to a diagonal D = UAV with d_1 \mid d_2 \mid \cdots. The invariant factors classify the abelian group \mathbb{Z}^m / A\mathbb{Z}^n: a factor 1 kills a generator, d &gt; 1 leaves a cyclic group \mathbb{Z}/d, and missing factors leave free \mathbb{Z} summands. The same reduction applied to a boundary matrix gives a space&#x27;s torsion.">

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  .. image:: /api/gallery/topology/homology/images/thumb/sphx_glr_plot_01_smith_normal_form_thumb.png
    :alt:

  :doc:`/api/gallery/topology/homology/plot_01_smith_normal_form`

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      <div class="sphx-glr-thumbnail-title">Smith's normal form of an integer matrix (1861)</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Enrico Betti counted the independent closed curves, surfaces, and so on, that do not bound in a space. In Poincaré&#x27;s later algebra these are the ranks \beta_k = \dim\ker\partial_k - \operatorname{rk}\partial_{k+1}: cycles minus boundaries. \beta_0 counts pieces, \beta_1 independent loops, and \beta_2 enclosed cavities. The bars show the bookkeeping for each space: chains f_k, cycles, and boundaries.">

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  .. image:: /api/gallery/topology/homology/images/thumb/sphx_glr_plot_02_betti_numbers_thumb.png
    :alt:

  :doc:`/api/gallery/topology/homology/plot_02_betti_numbers`

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      <div class="sphx-glr-thumbnail-title">Betti's connectivity numbers (1871)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Poincaré made Betti&#x27;s numbers a theory. Chains of simplices, their boundaries, and the homology groups H_k = \ker\partial_k / \operatorname{im}\partial_{k+1}, together with torsion coefficients. One consequence generalizes Euler: the alternating count of simplices equals the alternating sum of Betti numbers,">

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  .. image:: /api/gallery/topology/homology/images/thumb/sphx_glr_plot_03_poincare_homology_euler_formula_thumb.png
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  :doc:`/api/gallery/topology/homology/plot_03_poincare_homology_euler_formula`

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      <div class="sphx-glr-thumbnail-title">Poincaré's Analysis Situs: homology and the Euler-Poincaré formula (1895-1900)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Loops from a base point, up to deformation, form a group, \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 \langle a, b \mid aba^{-1}b^{-1}\rangle (up to naming), the projective plane \langle a \mid a^2\rangle, and abelianizing gives back H_1.">

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  .. image:: /api/gallery/topology/homology/images/thumb/sphx_glr_plot_04_fundamental_group_thumb.png
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  :doc:`/api/gallery/topology/homology/plot_04_fundamental_group`

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      <div class="sphx-glr-thumbnail-title">Poincaré's fundamental group (1895)</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/topology/homology/plot_01_smith_normal_form
   /api/gallery/topology/homology/plot_02_betti_numbers
   /api/gallery/topology/homology/plot_03_poincare_homology_euler_formula
   /api/gallery/topology/homology/plot_04_fundamental_group