# Thom class

An element in the (generalized) cohomology group of a 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) $E ^ {*}$, let $\gamma _ {n} \in \widetilde{E} {} ^ {n} ( S ^ {n} )$ be the image of $1 \in \widetilde{E} {} ^ {0} ( S ^ {0} )$ under the $n$-fold suspension isomorphism $\widetilde{E} {} ^ {0} ( S ^ {0} ) \cong \widetilde{E} {} ^ {n} ( S ^ {n} )$. Let $\xi$ be an $n$-dimensional vector bundle over a path-connected finite cell complex $X$, and let $j: S ^ {n} \rightarrow T ( \xi )$ be the corresponding inclusion into the Thom space. An element $u \in \widetilde{E} {} ^ {n} ( T)$ is called a Thom class (or orientation) of the bundle $\xi$ if $j ^ {*} u = \epsilon \gamma _ {n}$, with $\epsilon$ invertible in $\widetilde{E} {} ^ {0} ( S ^ {0} )$. A bundle need not have a Thom class. A bundle with a Thom class (in $E ^ {*}$) is called $E$-orientable, and a bundle with a fixed Thom class is $E$-oriented. The number of Thom classes of an $E$-orientable bundle over $X$ is equal to the number of elements of the group $( \widetilde{E} {} ^ {0} ( S ^ {0} )) ^ {*} \times \widetilde{E} {} ^ {0} ( X)$. Multiplication by a Thom class gives a Thom isomorphism.

For a (topological) manifold with or without boundary $( M , \partial M )$, a Thom class is a Thom class for its tangent (micro) bundle. Given a Thom class $t \in E ^ {n} ( M \times M , M \times M \setminus \Delta )$, there are isomorphisms $\varphi _ {t} : E _ {r} ( M \setminus B, M \setminus A) \widetilde \rightarrow E ^ {n- r} ( A, B)$ (Alexander duality), $E _ {r} ( A, B) \widetilde \rightarrow E ^ {n- r} ( M \setminus A, M \setminus B )$, $E _ {r} ( M, \partial M ) \widetilde \rightarrow E ^ {n- r} ( M)$ (Lefschetz duality) and $E _ {r} ( M) \widetilde \rightarrow E ^ {n- r} ( M, \partial M )$, $E _ {r} ( M) \rightarrow E ^ {n- r} ( M)$ (Poincaré duality), where $( M , \partial M )$ is a compact triangulable manifold and $B \subset A \subset M \setminus \partial M$ are compact subpolyhedra, cf. [a1], Chapt. 14, for more details.
An element $z \in E _ {n} ( M, \partial M)$ is called a fundamental class if for every $x \in M \setminus \partial M$ one has that $j _ {*} ( z) \in E _ {n} ( M, M \setminus \{ x \} )$ ($\cong E _ {n} ( U, U \setminus \{ x \} ) \simeq E _ {n} ( \mathbf R ^ {n} , \mathbf R ^ {n} \setminus \{ 0 \} )$) is a generator of $E _ {*} ( M, M \setminus \{ x \} )$ as a module over $E _ {*} ( pt)$. (Here $j$ is the inclusion $( M, \partial M ) \rightarrow ( M, M \setminus \{ x \} )$.) For the case of ordinary homology, cf. Fundamental class. The relation between a fundamental class and a Thom class is given by the result that if $M$ is a compact triangulable $n$-manifold with Thom class $t$, then there is a unique fundamental class $z \in E _ {n} ( M, \partial M )$ such that $\varphi _ {t} : E _ {n} ( M, \partial M ) \widetilde \rightarrow E ^ {0} ( M \setminus \partial M )$ takes $2$ to $1$, cf. [a1], Prop. 14.17. Using this the Lefschetz and Poincaré duality isomorphisms defined by the Thom class $t$ (which essentially are defined by a slant product with $t$) are given by a cap product with $z$.