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.

Comments

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 $.

References

[a1]	R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. Chapt. 2