View source for Fundamental class

The fundamental class of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421601.png" />-connected topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421602.png" /> (that is, a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421603.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421604.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421605.png" />) is the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421606.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421607.png" /> that corresponds, under the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421608.png" /> that arises in the universal coefficient formula

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to the inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216010.png" /> of the Hurewicz homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216011.png" /> (which is an isomorphism by the Hurewicz theorem (see [[Homotopy group|Homotopy group]])). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216012.png" /> is a [[CW-complex|CW-complex]] (a cellular space), then the fundamental class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216013.png" /> is the same as the first [[Obstruction|obstruction]] to the construction of a section of the [[Serre fibration|Serre fibration]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216014.png" />, which lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216015.png" />, and also as the first obstruction to the construction of a homotopy of the identity mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216016.png" /> to a constant mapping. In case the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216017.png" />-dimensional skeleton of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216018.png" /> consists of a single point (in fact this assumption involves no loss of generality, since any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216019.png" />-dimensional CW-complex is homotopy equivalent to a CW-complex without cells of positive dimension less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216020.png" />), the closure of each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216021.png" />-dimensional cell is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216022.png" />-dimensional sphere, and so its characteristic mapping determines some element of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216023.png" />. Since these cells form a basis of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216024.png" />, it thus determines an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216025.png" />-dimensional [[Cochain|cochain]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216026.png" />. This cochain is a [[Cocycle|cocycle]] and its cohomology class is also the fundamental class.

A fundamental class, or orientation class, of a connected oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216028.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216029.png" /> without boundary (respectively, with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216031.png" />) is a generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216032.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216033.png" /> (respectively, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216034.png" />), which is a free cyclic group. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216035.png" /> can be triangulated, then the fundamental class is the homology class of the cycle that is the sum of all coherent oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216036.png" />-dimensional simplices of an arbitrary triangulation of it. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216037.png" />, the homomorphism

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where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216039.png" />-product is defined by the formula

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is an isomorphism, called [[Poincaré duality|Poincaré duality]] (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216041.png" /> has boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216042.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216043.png" />). One also speaks of the fundamental class for non-oriented (but connected) manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216044.png" /> (with boundary); in this case one means by it the unique element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216045.png" /> (respectively, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216046.png" />) different from zero. In this case there is also a Poincaré duality.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.B. Fuks,   A.T. Fomenko,   V.L. Gutenmakher,   "Homotopic topology" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Mosher,   M.C. Tangora,   "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Husemoller,   "Fibre bundles" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.H. Spanier,   "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Dold,   "Lectures on algebraic topology" , Springer  (1980)</TD></TR></table>