View source for Thom class

An element in the (generalized) cohomology group of a [[Thom space|Thom space]], generating it as a module over the cohomology ring of the base space. For a multiplicative generalized cohomology theory (cf. [[Generalized cohomology theories|Generalized cohomology theories]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926601.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926602.png" /> be the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926603.png" /> under the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926604.png" />-fold [[Suspension|suspension]] isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926605.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926606.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926607.png" />-dimensional vector bundle over a path-connected finite cell complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926608.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t0926609.png" /> be the corresponding inclusion into the Thom space. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266010.png" /> is called a Thom class (or orientation) of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266011.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266012.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266013.png" /> invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266014.png" />. A bundle need not have a Thom class. A bundle with a Thom class (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266015.png" />) is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266017.png" />-orientable, and a bundle with a fixed Thom class is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266019.png" />-oriented. The number of Thom classes of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266020.png" />-orientable bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266021.png" /> is equal to the number of elements of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266022.png" />. Multiplication by a Thom class gives a [[Thom isomorphism|Thom isomorphism]].



====Comments====
For a (topological) manifold with or without boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266023.png" />, a Thom class is a Thom class for its tangent (micro) bundle. Given a Thom class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266024.png" />, there are isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266025.png" /> (Alexander duality), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266027.png" /> (Lefschetz duality) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266029.png" /> (Poincaré duality), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266030.png" /> is a compact triangulable manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266031.png" /> are compact subpolyhedra, cf. [[#References|[a1]]], Chapt. 14, for more details.

An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266032.png" /> is called a fundamental class if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266033.png" /> one has that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266034.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266035.png" />) is a generator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266036.png" /> as a module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266037.png" />. (Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266038.png" /> is the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266039.png" />.) For the case of ordinary homology, cf. [[Fundamental class|Fundamental class]]. The relation between a fundamental class and a Thom class is given by the result that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266040.png" /> is a compact triangulable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266041.png" />-manifold with Thom class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266042.png" />, then there is a unique fundamental class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266043.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266044.png" /> takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266045.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266046.png" />, cf. [[#References|[a1]]], Prop. 14.17. Using this the Lefschetz and Poincaré duality isomorphisms defined by the Thom class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266047.png" /> (which essentially are defined by a slant product with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266048.png" />) are given by a cap product with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092660/t09266049.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.M. Switzer,   "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. Chapt. 2</TD></TR></table>