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The complement of a closed tubular neighborhood of a smooth plane conic has the displayed integral homology. The Excision theorem and Thom isomorphism theorem reduce the calculation to the relative long exact sequence: the ambient degree-four fundamental class restricts with coefficient one, while the degree-two map is intersection with the conic and has coefficient two. A collar neighborhood identifies the open and compact exterior homotopy equivalence types. The boundary instead has first homology $\mathbb{Z}/4$, from the Gysin sequence and the normal Euler class four.

For this quotient, homotopy groups through degree six and in degree eight vanish. The degree-seven and degree-nine groups are $\mathbb{Z}/2$. A map to $K(\mathbb{Z}/2,7)$ induces an isomorphism on degree-nine integral homology because the relevant square of the bottom mod-two class is nonzero. Two relative Hurewicz steps then compute degrees eight and nine.

For maps between oriented connected closed manifolds of the same dimension, a homotopy induces equal maps on top-dimensional homology, hence equal coefficients on the fundamental class. For smooth maps and a smooth homotopy, the degree by integration of a pullback volume form gives another proof: the Generalized Stokes theorem on the homotopy cylinder makes the difference of endpoint pullback integrals zero because the target top form is closed. This is an obstruction to extending a degree-one sphere identity over a ball.

A homotopy sphere is a closed manifold with the homotopy type of a sphere of the same dimension. In dimensions at least two, a closed simply connected integral homology sphere is a homotopy sphere: successive applications of the Hurewicz theorem produce a map from the sphere inducing an isomorphism on integral homology, and the homological Whitehead theorem makes it a homotopy equivalence. The topological generalized Poincare theorem identifies its topological type, while prescribed smooth structures require a separate question.

Two spaces have the same homotopy type when there is a homotopy equivalence between them. Their homology and homotopy groups are then isomorphic. In Morse theory, attaching a cell records a change in the homotopy type of a sublevel set.

A Hopf band is an annulus with one full twist whose boundary is a Hopf link. Plumbing it with another band adds a generator to the first homology of a Seifert surface. Its core has self-linking number $1$ or $-1$, depending on the twisting convention.

Let $a,b\ge 2$ with $a+b=2n+1$. Collapse the lower cells of $S^a\times S^b$ to obtain $q:S^a\times S^b\to S^{2n+1}$ and compose with the Hopf fibration $p:S^{2n+1}\to {\mathbb{CP}}^n$. Both factor inclusions are killed by q, so every induced homotopy group map is zero; the induced reduced homology maps also vanish. If the composite were nullhomotopic, homotopy lifting property would deform q into the circle fibre. Every map from the simply connected product to that circle is nullhomotopic, contradicting the degree-one top homology map of q.

For an endomorphism of a finite-dimensional bounded chain complex over a field, the alternating chain trace equals the alternating trace on homology. Split each chain space into boundaries, representatives for homology, and a complement mapping isomorphically to the preceding boundaries. The chain-map relation makes the traces on the first and third summands cancel in successive degrees, leaving the homology traces. Over a field the induced cohomology maps are dual and have the same traces. This identity turns the absence of diagonal simplex contributions into a vanishing Lefschetz number.

For a simply connected space, if lower homotopy groups lie in an admissible Serre class, the Hurewicz map in the next degree is an isomorphism modulo that class, and the lower homology groups lie in the class. This applies to finitely generated groups and to finite groups supported at prescribed primes.

For $n\ge 1$, there is one integral cohomology generator $a_k$ in every degree $2nk$, and its products are $a_i{a}_j=\left(\genfrac{}{}{0ex}{}{i+j}{i}\right)a_{i+j}$. The Bott–Samelson theorem supplies free polynomial homology; its primitive generator gives the binomial diagonal dual to this divided power algebra.

Integral homology is homology with coefficients in $\mathbb{Z}$. It retains both the free and torsion parts of each homology group.

For $r\ge 3$, the path-loop Serre fibration has contractible total space. The relative homology of a fibration over a sphere gives $H_j(\Omega S^r)\cong H_{j-r+1}(\Omega S^r)$ for $j>0$, with negative homology zero and $H_0=\mathbb{Z}$. This proves the displayed groups. The unit map $S^{r-1}\to \Omega S^r$, adjoint to the identity of $S^r$, represents the generator of the first nonzero homotopy group. Naturality of the Hurewicz theorem shows that it induces an integral homology isomorphism in all degrees through $2r-3$.

Each real projective plane has cells $e_0,e_1,e_2$ with $d{e}_2=2e_1$ and $d{e}_1=0$. Their product cellular chain complex has ranks $1,2,3,2,1$. Its degree-two cycle $e_1\times e_1$ has order two in homology. The degree-three cycle $e_2\times e_1+e_1\times e_2$ also has order two, because its double is $d(e_2\times e_2)$. Consequently $H_0=\mathbb{Z}$, $H_1=(\mathbb{Z}/2{)}^2$, $H_2=H_3=\mathbb{Z}/2$ and all other groups vanish. The degree-three torsion is also the Tor functor term in the Künneth theorem.

For an oriented compact manifold of dimension $4k$, the intersection form is the bilinear pairing on middle-dimensional homology obtained by representing classes by transverse cycles and counting their signed intersection points. On an oriented four-manifold it is equivalently the cup-product pairing on $H^2$ evaluated on the fundamental class.

The intersection form of a 4-manifold is the four-dimensional case of this middle-dimensional intersection pairing.

Finite words in a based space, with occurrences of its basepoint deleted. Word concatenation defines a multiplication. For connected based CW spaces, the natural map to $\Omega \Sigma X$ is a weak homotopy equivalence. With free integral homology, the Bott–Samelson theorem identifies its homology algebra with the tensor algebra on reduced homology.

For $C_2\ast \mathbb{Z}\to S_3$ taking the two generators to a transposition and a $3$-cycle, the kernel's six-sheeted regular covering space consists of three spheres and two circles. Each sphere has two marked points, and each circle meets one point on each sphere. It retracts onto a rank-four graph by replacing each sphere by a meridian arc, but admits no deformation retraction onto any graph because its degree-two homology is ${\mathbb{Z}}^3$.

The Koszul complex on $f_1,\dots ,f_r$ has term $K_i={\bigwedge }^i{R}^r$ and differential $\partial (e_{j_1}\wedge \cdots \wedge e_{j_i})={\sum }_a(-1{)}^{a-1}{f}_{j_a}{e}_{j_1}\wedge \cdots \wedge \widehat{e_{j_a}}\wedge \cdots \wedge e_{j_i}$. Its degree-zero homology is $R/(f_1,\dots ,f_r)$. For a regular sequence it is exact in positive degrees, hence a Koszul resolution of that quotient. The variables of a polynomial ring form a regular sequence, so this resolves the coefficient field in length equal to the number of variables.

$L(f)={\sum }_i(-1{)}^i\mathrm{tr}(f_{\ast }:H_i\to H_i)$, equivalently the alternating trace on simplicial chains.

For an additive right-exact functor on module categories, apply it to a projective resolution and take homology. Lifts of module maps exist by projectivity; any two lifts are chain homotopic, giving independence and functoriality. The zeroth derived functor is naturally $F$, positive degrees vanish on projective modules, and the horseshoe lemma produces a long exact homology sequence from each short exact sequence. A Tor functor is the left derived functor of a tensor product of modules functor.

For a rational homology three-sphere $Y$, the linking form is the nonsingular pairingReversing the orientation of $Y$ negates this form.

Here the paired elements lie in the torsion subgroup of first homology.