A long exact sequence in (co)homology associated to an oriented sphere bundle (or more generally an oriented fiber bundle with appropriate fiber) that relates the cohomology of the total space, the base, and the fiber via pullback, pushforward (integration along the fiber, the Gysin map), and cup product with the Euler class.

Principle

Integration along the fiber (the Gysin or pushforward map) together with cup product by characteristic classes (notably the Euler class for sphere bundles) produces connecting morphisms that enforce exactness and relate degree shifts in cohomology between total space and base.

Demonstration

For an oriented S^n-bundle p: E → B, there is a long exact Gysin sequence ... → H^k(B) → H^{k+n+1}(B) → H^{k+n+1}(E) → H^{k+1}(B) → ... in which the connecting map is cup product with the Euler class and the pushforward appears as integration over the fiber; in the circle bundle case this relates to the first Chern class.

Misapplication

Using the Gysin sequence for bundles that are not oriented (without specifying twisted coefficients), applying it with incorrect degree shifts, or treating the Euler class as trivial when it is not leads to incorrect cohomological calculations.

Consequence

Provides a computable relationship allowing determination of the cohomology of E from that of B (and vice versa) when the characteristic classes and pushforward maps are known; it yields obstructions to bundle triviality and detects nontrivial classes coming from the fiber.

Reversal

If the bundle is trivial the Gysin sequence splits and the cohomology of the total space is the tensor product of base and fiber cohomology in the classical sense; reversing orientation or sign of the Euler class reverses certain connecting maps.

Boundary

Valid for oriented sphere bundles (or oriented fiber bundles with appropriate hypotheses) and within the chosen cohomology theory and coefficient system; it must be modified for non-orientable bundles (local coefficients), for generalized cohomology theories, or when base/fiber fail finiteness or orientability hypotheses.

Semantic Tension

Tension arises between using the Gysin sequence as a computational long exact sequence and using spectral sequences (e.g. Leray–Serre) which compute the same groups with different filtrations; each tool highlights different structures (exactness vs filtration and successive approximations).

Synthesis

The Gysin sequence organizes how cohomology classes of base and fiber combine in the total space via fiber integration and cup product with characteristic classes, giving exact algebraic relations that detect bundle nontriviality and compute cohomology across a fibration.