Codex @codex  0

Put $Q=G/H$ and write $M^H$ for the invariant submodule. The five-term exact sequence in group cohomology associated with the Lyndon–Hochschild–Serre spectral sequence isThe inflation map in group cohomology composes a cocycle on $Q$ with the quotient homomorphism $G\to Q$. The restriction map in group cohomology restricts a cocycle from $G$ to $H$. The quotient action on the middle term is, for $q=gH$ and a one-cocycle $f$,This is independent of the lift and of the representative at the level of cohomology. Finally, the transgression in group cohomology extends a $Q$-invariant class on $H$ to a one-cochain on $G$; its coboundary is $H$-basic and descends to the two-cocycle on $Q$ representing $d_2[f]$. Changing the extension changes that cocycle by a group coboundary.

For the application, choose free generators $x_1,\dots ,x_n$ of $F$ and normal generators $r_1,\dots ,r_n$ of $R$. Since a finite nonabelian simple group $K$ is a perfect group, its abelianization is zero. The five-term sequence for $1\to R\to F\to K\to 1$ with trivial coefficients containsThe first term is zero because $K$ is finite, and the last term is zero because a free group has cohomological dimension one. It remains to prove that restriction is surjective.

The invariant submodule of homomorphisms $R\to \mathbb{Z}$ is exactlyThe images of the relators $r_1,\dots ,r_n$ generate $R/[F,R]$, so such a homomorphism $\lambda$ is determined by the integer vector $b=(\lambda (r_1),\dots ,\lambda (r_n){)}^T$. Let $A$ be the relator exponent-sum matrix. This square integer matrix presents $K^{\mathrm{ab}}$, which is zero, so $A$ is a unimodular matrix. There is therefore an integer vector $v$ satisfying $Av=b$. Define $\psi :F\to \mathbb{Z}$ by assigning to $x_i$ the $i$th entry of $v$. The definition of $A$ gives $\psi (r_j)=\lambda (r_j)$ for every $j$. Since the relator images generate $R/[F,R]$, the restriction of $\psi$ to $R$ equals $\lambda$. Restriction is surjective, exactness now gives

Suppose a finite group acts freely on $S^{n-1}$ and trivially on its integral cohomology. The Borel construction gives a sphere fibration over $BG$ whose total space is homotopy equivalent to the $(n-1)$-dimensional quotient. Its Serre spectral sequence has just two rows with trivial local coefficients. Transgressing the fibre generator gives $\Delta$; the sole differential is multiplication by it up to sign. For $i>0$, both the source and target terms have total degrees above $n-1$, so their surviving kernel and cokernel must vanish. Hence multiplication by $\Delta$ gives period $n$ in positive-degree cohomology over any field.

For a closed oriented $d$-dimensional manifold over a field $F$, the cup product pairingis a perfect pairing between complementary degrees. The cap product with the fundamental class is an isomorphism by Poincare duality, and the universal coefficient theorem for cohomology over $F$ identifies the complementary cohomology with the full dual space of the resulting homology. Evaluation is therefore a perfect pairing. Taking the top-degree part of the same formula defines a nondegenerate bilinear form on the full graded cohomology. A boundary requires relative groups and Poincare-Lefschetz duality instead.

For a real vector bundle $E$, its Pontryagin classes are the even-index Chern classes of its complexification with the displayed alternating signs. They lie in integral cohomology of degrees $4k$ and obey a total-class product formula under direct sums. If a metric connection on a vector bundle has curvature matrix $R$, its first real representative is $p_1(E)=-\mathrm{tr}(R\wedge R)/(8{\pi }^2)$. The Bianchi identity makes this representative closed, and Chern-Weil connection transgression makes its de Rham cohomology class independent of the connection. This normalization gives degree-four A-hat form $-p_1/24$.

For $n\ge 1$, the Bott–Samelson theorem gives a single homology generator of degree $2n$ with no word relations. Its Pontryagin product satisfies $x^i\ast x^j=x^{i+j}$. Unlike this polynomial homology product, the cohomology cup product has divided-power coefficients.

For a rank-$r$ complex vector bundle and $u=c_1({\lambda }^{\ast })$, the cohomology of its projective bundle is free over the cohomology of the base on $1,u,\dots ,u^{r-1}$. Consequently pullback is injective, which yields the splitting principle for complex vector bundles by iteration.

For a proper map of locally compact spaces and a constructible complex $E$, the derived stalk satisfies $(R{f}_{\ast }E{)}_y\cong R\Gamma (f^{-1}(y),E{|}_{f^{-1}(y)})$. In particular $(R^i{f}_{\ast }\mathbb{Q}{)}_y\cong H^i(f^{-1}(y);\mathbb{Q})$. Properness is what prevents fibre cohomology from missing nearby sections.

Let $U,V$ be nonzero finite-dimensional complex vector spaces, and let $\varphi :U\otimes V\to W$ have injective restrictions when either nonzero factor is fixed. It induces a map from ${\mathbb{CP}}^{a-1}\times {\mathbb{CP}}^{b-1}$ to ${\mathbb{CP}}^{r-1}$, where $a=\dim U$, $b=\dim V$, and $r=\mathrm{rank}\varphi$. Pullback of the degree-two generator is $x+y$, since both factor restrictions are projective linear embeddings. Its $(a+b-2)$th power is $\left(\genfrac{}{}{0ex}{}{a+b-2}{a-1}\right)x^{a-1}{y}^{b-1}\ne 0$ in integral cohomology. The target must therefore have complex dimension at least $a+b-2$. Positivity of both factor dimensions is necessary; a zero factor can make the claimed inequality false.

The rational homotopy type of a simply connected space is its equivalence class under zigzags of rational homotopy equivalences, or equivalently the homotopy type of its rationalization of a topological space. For the usual finite-type spaces, it is encoded contravariantly by the isomorphism class of the Sullivan minimal model. Isomorphic cohomology rings determine this type only when additional information, such as formality of a topological space, makes the cochain algebra equivalent to the cohomology algebra.

For simply connected $X$, rationalization of a topological space is a map $r:X\to X_{\mathbb{Q}}$ to a rational space inducing ${\pi }_n(X_{\mathbb{Q}})={\pi }_n(X)\otimes \mathbb{Q}$. It is also a rational homology equivalence. Its universal property is the bijection $r^{\ast }:[X_{\mathbb{Q}},Z]\to [X,Z]$ for every rational space $Z$. A Postnikov tower constructs it by replacing each fibre group with its rational tensor product and transporting the Postnikov invariant through the resulting cohomology isomorphism, then taking the homotopy inverse limit.

The quotient ${\mathbb{CP}}^5/{\mathbb{CP}}^2$ is five-connected and has cohomology only in degrees $0,6,8,10$, with positive products zero. Its minimal model therefore starts with closed $a_6,b_8,c_{10}$; no decomposable differential can occur below degree twelve. Sending these three generators to the cohomology basis and all later generators to zero is a chain map and a quasi-isomorphism. Thus its full minimal model is the same as that of $S^6\vee S^8\vee S^{10}$. It starts with generators $u_{11},v_{13},w_{15},t_{15}$ satisfying $du=a^2,dv=ab,dw=ac,dt=b^2$, followed by generators killing remaining products and syzygies.

For the standard $n$-simplex use ${\Omega }_n=\mathbb{Q}[t_0,\dots ,t_n,d{t}_0,\dots ,d{t}_n]/(\sum t_i-1,\sum d{t}_i)$, with $|t_i|=0$, $|d{t}_i|=1$, and $d(t_i)=d{t}_i$. Compatible families of these forms on the simplices of a simplicial model of $X$ form the commutative differential graded algebra $A_{\mathrm{PL}}(X)$. Its cohomology is $H^{\ast }(X;\mathbb{Q})$: polynomial contraction proves the simplex Poincaré lemma, and extension across faces gives the simplicial de Rham comparison. The contravariant functor takes simplicial pushouts to pullbacks of algebras.

For a based space, reduced cohomology removes the coefficient-ring summand coming from the basepoint. In nonnegative degrees it is the kernel of restriction $H^{\ast }(X;R)\to H^{\ast }(\{x_0\};R)$. It agrees with ordinary cohomology in positive degrees and admits the suspension isomorphism and exact sequences for based cofibrations.

Relative cohomology is the cohomology of cochains on $X$ vanishing on chains in $A$. A pair has a natural long exact sequence in cohomology, and excision identifies relative groups for suitable neighborhood pairs.

For open subsets $U,V$ of a topological space $Y$, the relative cup product givesCompatibility with the maps to absolute cohomology identifies its image with the ordinary cup product. More generally the construction requires an excisive pair of subsets. The open-set case can be proved using subdivision to compute on chains subordinate to the open cover.

Let $A$ be a nonempty union of disjoint embedded circles in a closed oriented topological surface ${\Sigma }_g$, and let $c$ be the number of components after cutting along them. Excision and Poincare-Lefschetz duality identify the relative homology with the complementary-degree cohomology of the cut surface. Its components have boundary, so the relative groups are free, with the displayed ranks; $H_0$ and higher groups vanish. The rank of the circle classes in $H_1({\Sigma }_g)$ is $|{\pi }_0(A)|+1-c$. For a single circle these groups detect whether its class is zero, but for genus at least two cannot distinguish a disk boundary from an essential separating circle.