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Start with a finite $n$-dimensional $n$-connected approximation. Its degree-$n$ homology kernel is finite free. Realize a basis via the Relative Hurewicz theorem and attach $(n+1)$-cells; their boundary is injective, so they kill precisely that kernel without adding top homology. The homological Whitehead theorem gives a finite weak equivalence of dimension at most $n+1$.

For simply connected $X$ with finitely generated integral homology in every degree, the Hurewicz theorem modulo a Serre class gives finitely generated homotopy groups. Realize finite generating sets of successive relative homotopy groups by attaching finitely many cells at each stage. The resulting weak equivalence has finitely many cells in each dimension.

For a compact connected Riemannian manifold with nonnegative Ricci curvature, its finitely generated fundamental group acts freely and properly discontinuously on the complete universal cover. The preceding polynomial growth of properly discontinuous isometry groups bound applies. The degree-one Hurewicz theorem identifies first integral homology with the abelianization of the fundamental group. If its free rank is $b_1(M)$, this group maps onto ${\mathbb{Z}}^{b_1(M)}$. Choosing lifts of a basis gives at least a constant times $m^{b_1(M)}$ distinct elements in a word ball of radius a constant times $m$. The upper bound of order $m^{\dim M}$ forces the displayed inequality.

Compatible orientations of all top-dimensional simplices make their signed sum a cycle. For a connected pseudomanifold this cycle generates top homology over the integers; if no compatible orientation exists, top integral homology is zero.

A graded Hom functor construction consists of maps shifting degree by $j$. One homological convention is $d_{M}f=f{d}_C+(-1{)}^{j-1}{d}_{C^{\prime }}f$. Its degree-zero cycles are chain maps, and its degree-zero boundaries are null-homotopic chain maps, so its zeroth homology is the module of chain homotopy classes. This convention differs by degree-dependent signs from the alternative convention $d^{\prime }f-(-1{)}^{j}fd$.

Group homology is obtained by tensoring a projective resolution of the trivial $\mathbb{Z}G$-module with a coefficient module and taking homology.

The Haken sum of compatible normal surfaces is obtained by resolving their intersection curves in the way that preserves the normal disk types. It adds normal surface coordinates, Euler characteristic and mod-two homology classes. The alternative local resolution can create returning face arcs and need not be normal.

Leonhard Euler's Elementa doctrinae solidorum, or Elements of the doctrine of solids, appeared in 1758 in volume 4 of Novi Commentarii academiae scientiarum Petropolitanae, pp. 109–140. The repository record dates its composition to 1750. Composition and publication are distinct dates here.^[ref]

The modern cell-count formula for Euler characteristic puts polyhedral counting into an arbitrary-dimensional CW complex. It also identifies what is being counted: the Euler characteristic of a cube boundary and its interior distinguishes a polyhedral surface from the filled solid using the missing three-dimensional term. The modern homology formulation explains why this alternating arithmetic persists across different decompositions.^[ref]

The Hochschild homology of a $k$-algebra with coefficients in a bimodule is the homology

Homological algebra studies chain complexes, homology, cohomology, and the derived constructions that measure failure of exactness.

A smooth compact Riemann surface embedded in Complex projective space ${\mathbb{CP}}^2$ determines an integral homology class $d[{\mathbb{CP}}^1]$. Since a projective line has self-intersection one, $d$ is the intersection number with a transverse projective line. Every transverse intersection of complex curves has positive sign, because a complex-linear change of tangent basis has positive real determinant. For the Fermat curve $x^d+y^d=z^d$, the line $y=0$ meets it transversely in the $d$ points $[\zeta :0:1]$, ${\zeta }^d=1$, directly verifying its homological degree.

Homological monodromy is the induced automorphism of the homology of a fiber. For a fibered knot, its action on the first homology group presents the Alexander module.

For an $n$-sheeted covering space $f:X\to Y$, define the transfer of a singular simplex as the sum of its $n$ lifts. Restriction of these lifts to any face gives exactly the lifts of that face, so transfer is a chain map. Its induced map $\mathrm{tr}:H_i(Y;R)\to H_i(X;R)$ satisfies $f_{\ast }\mathrm{tr}=n\mathrm{id}$ for any commutative coefficient ring $R$. If $n$ is invertible in $R$, then $n^{-1}\mathrm{tr}$ is a right inverse of $f_{\ast }$. In particular a finite covering surjects onto the base's rational homology.

An integral homology equivalence between simply connected spaces is a weak homotopy equivalence. A first nonzero relative homotopy group would contradict the Relative Hurewicz theorem. Between CW complexes the ordinary Whitehead theorem then gives a homotopy equivalence.

Homology assigns abelian groups $H_i(X)$ to a space, measuring cycles modulo boundaries in each dimension.

Glue the boundary of a Möbius band to the projective line in the real projective plane by a mapping degree $n$. The band is the mapping cylinder of the degree-two boundary covering of its core. Collapsing the connecting tree in the resulting double mapping cylinder gives a CW complex with one vertex, edges $a,b$, and two attaching words $a^2$ and $a^{-n}{b}^2$. Its cellular chain complex has degree-two boundary matrix $\left(\begin{array}{cc}2& -n\\ 0& 2\end{array}\right)$. Its determinant is four, so $H_2=0$. The Smith normal form gives $H_1\cong \mathbb{Z}/4$ for odd $n$ and $(\mathbb{Z}/2{)}^2$ for even $n$; $H_0=\mathbb{Z}$ and higher homology vanishes. The space cannot be homotopy equivalent to a closed manifold: mod-two homology forces a potential manifold to be a surface, and its Euler characteristic is one, whereas the only such closed connected surface is the projective plane, with different integral $H_1$.

The integral homology of the loop space of a sphere implies that the unit $S^{r-1}\to \Omega S^r$ has vanishing relative homology through degree $2r-3$. Both spaces are simply connected for $r\ge 3$. The absolute Hurewicz theorem first identifies their degree-two homotopy groups with homology; the relative Hurewicz theorem then rules out a first nonzero relative homotopy group in degrees three through $2r-3$. The relative homotopy sequence makes the unit an isomorphism on ${\pi }_i$ through degree $2r-4$. Loop-suspension adjunction identifies this map with suspension, yielding ${\pi }_i(S^{r-1})\cong {\pi }_{i+1}(S^r)$ in that range.

The singular-chain product induces this natural pairing on homology. Composing it with multiplication of a loop space gives the Pontryagin product on loop-space homology. This is different from the vector cross product.

A homology group is one of the abelian groups produced by a homology theory. Its elements are homology classes, represented in singular homology by cycles modulo boundaries.

A homotopy equivalence of a closed surface cannot carry a simple closed curve of nonzero integral homology class onto a contractible simple closed curve. Any map into the latter factors through a circle whose inclusion induces zero on first homology; the former class would be killed by an isomorphism. Such a carrying map can nevertheless exist: project a handle generator onto a circle and compose with the contractible embedded circle.