.. _Section 2.1:

2.1. Simplicial and Singular Homology
======================================

|indent| The most important homology theory in algebraic topology, and the one we shall
be studying almost exclusively, is called singular homology. Since the technical apparatus
of singular homology is somewhat complicated, we will first introduce a more
primitive version called simplicial homology in order to see how some of the apparatus
works in a simpler setting before beginning the general theory.

|indent| The natural domain of definition for simplicial homology is a class of spaces we
call :math:`\Delta`-complexes, which are a milde generalization of the more classical notion of
a simplicial complex. Historically, the modern definition of singular homology was
first given in [Eilenberg 1944], and :math:`\Delta`-complexes were introduced soon thereafter in
[Eilenberg-Zilber 1950] where they were called semisimplicial complexes. Within a
few years this term came to be applied to what Eilenberg and Zilber called complete
semisimplicial complexes, and later there was yet another shift in terminology as 
the latter objects came to be called simplicial sets. In theory this frees up the term
semisimplicial complex to have its original meaning, but to avoid potential confusion
it seems best to introduce a new name, and the term :math:`\Delta`-complex has at least the virtue
of brevity.

.. toctree::
    delta-complexes
    simplicial-homology
    singular-homology
    homotopy-invariance
    exact-sequences-and-excision
    the-equivalence-of-simplicial-and-singular-homotopy

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