Generalized cohomology theory ($h^n(X,A)$)

A generalized cohomology theory assigns graded groups contravariantly to suitable topological pairs, with homotopy invariance, excision and a natural long exact sequence for each pair. It omits the dimension axiom of ordinary cohomology, so its coefficient groups at a point can occur in degrees other than zero. A multiplicative theory also has compatible external and relative cup products.

A topological spectrum $E$ represents the reduced generalized cohomology theory ${\tilde{E}}^q(X)=[{\Sigma }^{\infty }X,{\Sigma }^{q}E{]}_{\mathrm{st}}$. For an Omega-spectrum this equals $[X,E_q{]}_{\ast }$, taking $E_q={\Omega }^{-q}{E}_0$ for negative $q$. It is contravariant in $X$.

A generalized cohomology theory with natural unital cup products compatible with relative groups, suspension, and the cohomology axioms. Its coefficient groups form a graded ring. Thom orientations make the cohomology of a bundle's Thom space a shifted module over the cohomology of the base.

// Target: geometry-and-topology.bigb

The skeletal filtration of a CW complex gives this spectral sequence, with differential of bidegree $(r,1-r)$ in cohomological grading. For a finite complex the filtration is finite and convergence is bounded. In a multiplicative theory the associated graded product is the ordinary cohomology product with coefficients in the theory's coefficient ring.

The skeletal filtration of a CW complex and the cofiber sequences of spectra of its cell attachments produceFor a connective spectrum it is first quadrant and converges in each fixed total degree. Its reduced version has no contribution from the basepoint summand.