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Let $M$ be a closed Riemannian manifold and $E$ a flat Hermitian vector bundle with unitary holonomy. On $E$-valued differential forms of degree $q$, let ${\Delta }_q$ be the nonnegative Hodge Laplacian. For its positive eigenvalues, defineusing analytic continuation to $s=0$; the prime omits zero eigenvalues. The Ray-Singer analytic torsion isThis packages the form spectra into a regularized determinant, supplying an analytic counterpart of Reidemeister torsion.^[ref]

For acyclic unitary coefficients on a closed manifold, this scalar is independent of the Riemannian metric. With nonzero cohomology, the scalar alone can depend on the metric; the full construction also tracks the norm of cohomology bases. For a circle of length $L$ with trivial coefficients, the positive eigenvalues are $(2\pi k/L{)}^2$ for $k\ne 0$ in degrees zero and one. Their zeta regularization gives $T=L$, illustrating that dependence.^[ref]

Turaev torsion refines the combinatorial side's sign and cellular-lift choices; comparing analytic and combinatorial constructions requires matching those conventions.

The Atiyah-Bott fixed-point theorem computes an alternating trace on the cohomology of an elliptic complex from local data at fixed points. In the isolated, nondegenerate case, let $f:M\to M$ be a smooth map of a closed smooth manifold, with compatible lifts ${\varphi }_i:f^{\ast }{E}_i\to E_i$ commuting with the complex's differential operators. If $\det (1-d{f}_x)\ne 0$ at every fixed point, thenThe lifts, not just the map of the base, determine the action.^[ref]

For the de Rham complex and reflection of a circle, there are two fixed points with derivative $-1$. At each, the alternating fibre trace is $1-(-1)=2$ and the denominator is two, giving local contribution one. Globally the induced actions on $H^0$ and $H^1$ are $1$ and $-1$, so both sides equal two. The holomorphic Lefschetz fixed-point formula applies the same idea to holomorphic complexes.

The degree-one Bockstein homomorphism makes modulo-$n$ cohomology into a cochain complex. Its cohomology is the Bockstein cohomology. For prime $n$ this is a graded vector space; for composite $n$ it is a graded $\mathbb{Z}/n$-module. For ${\mathbb{RP}}^3$ at modulus two, its only nonzero groups are ${\mathbb{F}}_2$ in degrees zero and three. This is the cohomological counterpart of Bockstein homology.

For a CW complex $X$ and coefficient abelian group $A$, the cohomology of its cellular cochain complex naturally equals its singular cohomology:The cellular cochain group is $\mathrm{Hom}(C_q^{\mathrm{cell}}(X;\mathbb{Z}),A)$, and its coboundary map is precomposition with the cellular boundary.

If $X$ has only even-dimensional cells, every cellular coboundary map is zero. If, in addition, there are finitely many cells in each dimension, then its integral cohomology groups are free abelian groups, with one generator for each cell in that degree. For infinitely many cells in one degree, the cochain group is a direct product of groups of copies of $\mathbb{Z}$, rather than the direct sum used for cellular chains; the finite-cell freeness conclusion must not be applied without that qualification. Even with all additive groups known, the cup product still requires a separate computation.

For a positively graded Lie algebra $L$ over $\mathbb{Q}$, finite-dimensional in each degree, its cochain algebra is free graded commutative on the graded dual of $sL$, with differential dual to the Lie bracket and its Koszul signs. The Jacobi identity gives $d^2=0$. A Lie element of degree $r$ contributes a cochain generator of degree $r+1$. Its cohomology computes ${\mathrm{Ext}}_{U(L)}(\mathbb{Q},\mathbb{Q})$ with total grading; the standard free enveloping-algebra resolution gives this identification. For a free graded Lie algebra, the tensor-algebra augmentation has a length-one free resolution, explaining the square-zero cohomology of a rational wedge sum of spheres.

A cochain differential is a degree-one map in a cochain complex:The square-zero identity makes every coboundary a cocycle, so the quotient $H^n(C)=\ker d^n/\mathrm{im}{d}^{n-1}$ defining cohomology is meaningful. The maps are homomorphisms of the abelian groups, or module-linear maps, used in the complex.

For $C^0=\mathbb{Z}$, $C^1=\mathbb{Z}$, $d^0(n)=2n$, and all other groups zero, the identity is automatic. The degree-zero cohomology is zero, while the degree-one cohomology is $\mathbb{Z}/2\mathbb{Z}$: every degree-one element is closed, but only the even ones are exact.

The exterior derivative on differential forms is another cochain differential, with $d^2=0$. In contrast, a chain differential lowers degree. The word differential records this graded square-zero structure, not necessarily differentiation of scalar functions.

For cochain maps $f,g:C^{\bullet }\to D^{\bullet }$, a cochain homotopy is a family $K:C^q\to D^{q-1}$ satisfyingCochain-homotopic maps induce the same map on cohomology.

A cochain map $f:E\to F$ consists of degreewise maps $f^i:E^i\to F^i$ satisfying $d_F^i{f}^i=f^{i+1}{d}_E^i$. This identity sends cocycles to cocycles and coboundaries to coboundaries, hence induces maps on cohomology. For complexes of sheaves the same identity holds as an identity of sheaf morphisms.

For a map between closed oriented manifolds of the same dimension, Poincare duality turns homological pushforward into a degree-preserving map on cohomology. It is adjoint to pullback under the duality pairing: $⟨f^{!}a⌣b,[N]⟩=⟨a⌣f^{\ast }b,[M]⟩$. If $f$ has degree $d$, nondegeneracy gives $f^{!}{f}^{\ast }=d\mathrm{id}$. Maps of different dimensions require the corresponding degree shift.

A cohomology theory assigns abelian groups $H^q(X;R)$ to a topological space $X$. For singular cohomology with coefficients in a commutative ring $R$, form the cochain complex $C^q(X;R)$ with coboundary $\delta$ and takeA cocycle represents a cohomology class; adding a coboundary leaves that class unchanged.

A cohomology class evaluates on a homology class by applying its cochain to a representing cycle. Replacing either representative changes the result by zero, explaining the relation to homology. A continuous map $f:X\to Y$ induces cochain pullback and hence $f^{\ast }:H^q(Y;R)\to H^q(X;R)$: the direction reverses.

The cup product combines degree-$p$ and degree-$q$ classes into degree $p+q$, giving a graded cohomology ring. The circle has one integral generator in degrees zero and one. A two-dimensional torus has two independent degree-one classes whose cup product generates degree two. That multiplication distinguishes it from $S^1\vee S^1\vee S^2$, which has the same additive cohomology groups but zero products between its positive-degree generators.

An oriented normal bundle of a compact embedded submanifold has a Thom class. A tubular neighborhood and excision identify it with a relative class in $H^d(X,X\setminus Y)$. Its image in absolute cohomology is ${\epsilon }_Y$. It restricts to zero off Y and, in an oriented closed ambient manifold, represents the Poincare dual of Y with the normal-first induced orientation.

For the Hopf map $\eta$, the indicated four-cell attachment has cellular boundary multiplication by $f$ from dimension four to three. Its integral cohomology is $\mathbb{Z}$ in degrees zero and two, $\ker (f:\mathbb{Z}\to \mathbb{Z})$ in degree three, and $\mathbb{Z}/f\mathbb{Z}$ in degree four. If $x$ generates degree two and $z$ is the four-cell cochain class, then $x^2=dz$, with every other positive-degree product zero. Collapsing the three-sphere reduces the cup-square calculation to the Hopf invariant; the induced map on degree-four cohomology reduces its integer coefficient modulo $f$. When $f=0$ there is an extra free degree-three class, whose products still vanish by dimension.

For a cochain complex of sheaves $E$, its $i$th cohomology sheaf is ${\mathcal{H}}^i(E)=\ker (d^i)/\mathrm{im}(d^{i-1})$, where the image and quotient are taken in sheaves. Its stalk at $x$ is the cohomology of the stalk complex, because taking stalks is exact.

A short exact sequence of cochain complexes of sheaves $0\to E\to F\to G\to 0$ induces $\cdots \to {\mathcal{H}}^i(E)\to {\mathcal{H}}^i(F)\to {\mathcal{H}}^i(G)\to {\mathcal{H}}^{i+1}(E)\to \cdots$. Taking stalks is exact, so the construction and exactness follow from the ordinary long exact sequence of cohomology of complexes at every stalk. The stalkwise connecting maps come from the lift-and-differentiate construction and form a sheaf morphism.

Pull back along loop evaluation $S^1\times \Omega B\to B$ and take the slant product with the circle's fundamental class. This degree-lowering operation kills positive-degree cup products. It commutes with stable cohomology operations such as Steenrod squares and identifies a transgressing base class with its fiber class in the path-loop fibration.

A compact Lagrangian submanifold $L$ with vanishing first de Rham cohomology cannot be a fibre of a smooth positive-dimensional fiber bundle whose nearby fibres are also Lagrangian submanifolds. In a Weinstein neighborhood, nearby fibres are one-form graphs of closed forms. They are exact by the cohomology assumption, so meet the zero section at a critical point of their potential, contradicting disjointness of distinct fibres.

A compact smooth $n$-manifold with nonempty boundary embedded in $S^n$ is a proper compact locally contractible subset. Alexander duality therefore expresses the reduced homology of its complement by the reduced cohomology of the abstract manifold. Thus these groups do not depend on its particular smooth embedding.

For connected oriented closed $d$-dimensional manifolds with $d\ge 2$, their connected sum removes an open $d$-dimensional ball from each and identifies the resulting boundary spheres by an orientation-reversing diffeomorphism. The remaining orientations fit together. Collapsing the separating sphere gives a pinch map to the wedge sum of the two closed manifolds, and projection to either summand has degree of a continuous mapping one. In intermediate positive degrees the cohomology is the direct sum of the summand groups. Products of classes from different summands vanish; products landing in top degree use the single common orientation class. These statements follow from excision, the Mayer–Vietoris sequence and the degree-one projections.

The derived costalk at $x$ is $i_x^{!}E$, where $i_x:\{x\}\hookrightarrow X$. It computes local cohomology supported at $x$. On an oriented $N$-manifold the constant sheaf has costalk $\mathbb{Q}[-N]$, whose cohomology is in degree $N$.

Let $M,N$ be closed connected oriented manifolds of the same dimension, and suppose $a^r$ generates the top integral cohomology of $N$. If every class of degree $|a|$ on $M$ has zero rth cup power, every continuous map $f:M\to N$ has mapping degree zero. Indeed, $f^{\ast }(a^r)=(f^{\ast }a{)}^r=0$, and evaluation on the fundamental class gives $\mathrm{deg}f=0$. For $S^2\times S^4\to {\mathbb{CP}}^3$, the degree-two class is a multiple of the sphere class $u$, whose square vanishes.