.. _sphx_glr_api_gallery_topology_complexes:

Complexes
---------

Constructions on simplicial complexes: subdivision, products, and gluing.


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    <div class="sphx-glr-thumbcontainer" tooltip="Is homology a property of the space or of the chosen triangulation? Brouwer&#x27;s simplicial approximation (1911) answers it. Any continuous map is close to a simplicial one after subdividing finely enough. Alexander (1915) used this to prove that homology is a topological invariant. Barycentric subdivision replaces every simplex by the chains of its faces. The simplex counts grow about sixfold per round for surfaces, while the Betti numbers do not change.">

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  .. image:: /api/gallery/topology/complexes/images/thumb/sphx_glr_plot_01_barycentric_subdivision_invariance_thumb.png
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  :doc:`/api/gallery/topology/complexes/plot_01_barycentric_subdivision_invariance`

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      <div class="sphx-glr-thumbnail-title">Brouwer, Alexander, and the invariance of homology under subdivision (1911-1915)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Hermann Künneth showed that the Betti numbers of a product are a convolution: \beta_k(X \times Y) = \sum_{i+j=k} \beta_i(X)\,\beta_j(Y), the coefficients of the product of the two Poincaré polynomials. The product of two simplicial complexes is triangulated by staircase paths through each product of simplices. Circle times circle is the torus, (1 + t)^2 = 1 + 2t + t^2, and circle times sphere gives (1 + t)(1 + t^2).">

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  :doc:`/api/gallery/topology/complexes/plot_02_kunneth_formula`

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      <div class="sphx-glr-thumbnail-title">The Künneth formula for products (1923)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Walther Mayer and Leopold Vietoris computed the homology of a union from its pieces through the long exact sequence">

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  .. image:: /api/gallery/topology/complexes/images/thumb/sphx_glr_plot_03_mayer_vietoris_sequence_thumb.png
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  :doc:`/api/gallery/topology/complexes/plot_03_mayer_vietoris_sequence`

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      <div class="sphx-glr-thumbnail-title">The Mayer-Vietoris sequence (1929-1930)</div>
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.. toctree::
   :hidden:

   /api/gallery/topology/complexes/plot_01_barycentric_subdivision_invariance
   /api/gallery/topology/complexes/plot_02_kunneth_formula
   /api/gallery/topology/complexes/plot_03_mayer_vietoris_sequence