Orientation transport
نویسنده
چکیده
Let (Y, g) denote a compact, oriented Riemann 3-manifold without boundary. Denote by π : X → Y a principal S1-bundle over Y , and by Z → Y the associated 2-disk bundle. Set c := c1(Z) ∈ H(Y,Z). Denote by tZ ∈ H2(Z, X;Z) the Thom class of Z → Y , by j the inclusion X ↪→ Z and by ζ : Y ↪→ Z the natural inclusion. Using the Thom isomorphism H•(Z) ∪tZ −→ H•+2(Z,X;Z), c = ζ∗tZ , and the long exact cohomological sequence of the pair (Z, X) we obtain the Gysin sequence · · · π! −→ Hk−2(Y,Z) ∪c −→ H(Y,Z) π∗ −→ H(X,Z) π! −→ Hk−1(Y,Z) ∪c −→ · · ·
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