The intuition of Lefschetz hyperplane theorem comes from its proof using Morse theory, in particular Andreotti–Frankel theorem, saying that an smooth, complex affine variety of complex dimension has homotopy type of a (real) -dimensional CW complex (This implies vanishing of cohomology for affine complex variety above .). The statement then follows by applying to and Alexander-Lefschetz duality. The idea is to use excision applied to a ANR (to be able to use excision) together with Poincare duality. See Hatcher, theorem 3.44. See the derivation here.
Cohomology with compact support: . They are the set of cochains which vanishes outside a compact set .
Relationship with Gysin sequence: The idea is that Thom space gives rise to the Euler class, which is the image of the orientation class in under the pullback . The important thing about Euler class is that it is the vanishing locus of generic section and the Euler class of the normal bundle of in is naturally identified with the self-intersection of in .
Stable characteristic class:
Stiefel–Whitney class (a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of sections of the vector bundle. Stiefel–Whitney classes are indexed from to , where is the rank of the vector bundle)
Chern class (complex analogue of Stiefel-Whitney class)
See this thread for why there is no curvature form interpretation of Stiefel-Whitney class.
Connection to Steenrod square: It is the algebra of stable cohomology operations for mod cohomology, see this note for an introduction.
More on Steenrod square: One perspective is that Steenrod squares remember normal bundle data (self-intersection), see this answer. Another perspective is that it measures how the cup product, while homotopy-commutative (in terms of the induced maps to Eilenberg-MacLane spaces), cannot be straightened to be actually commutative, see this answer for an explanation. For more detail see Hatcher, page 502 and this note. The idea is that a cohomology class has cup product , which can be viewed as a map of . We can extend the last map to by virtue of the homological commutativity of cup product (which translates to existence of homotopy from to where is the self-map of swapping the two coordinates and since we get a loop of maps . After choosing appropriate this map will be null-homotopic so extends to and we iterate this process.). This map has the property that and has the same image, so it descends to which extends . Now we use Kunneth formula and write as where is the generator of . The is defined to be .
The intuition of Adem relations is that they come from the symmetry of by swapping the factor. They actually follow from other axioms of Steenrod square, see this paper and this short note. For more see here.
The Adem relations follows from two facts: 1. 2. is a derivation. Then we have a Pascal’s triangle. See this note.
Previously we have computed the cohomology ring of . With integral information, we should look at each prime. Serre showed that . See here and Hatcher’s Spectral Sequence, Section 5.1 for a proof.
The theorem implies that the admissible monomials in are linearly independent, hence form a basis for as a vector space over . For if some linear combination of admissible monomials were zero, then it would be zero when applied to the class , but if we choose larger than the excess of each monomial in the linear combination, this would contradict the freeness of the algebra.
Another consequence is that a cohomological operation commutes with suspension (i.e. the stable cohomological operation) iff it is of the form for some . The same holds if we replace ‘stable’ by ‘linear’ (note that a cohomological operation need not be a homomorphism).
Application of Steenrod square: If has Hopf invariant 1, then is nonzero (Theorem 4L.2 of Hatcher).
The Adem relations implies is generated by as an algebra by the elements .