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Give a discrete group $G$ a contractible free CW complex $EG$. The quotient $BG=EG/G$ has universal cover $EG$, so it has fundamental group $G$ and no higher homotopy groups. It is thus an Eilenberg–MacLane space $K(G,1)$, even when $G$ is nonabelian. The cohomology of $BG$ with the appropriate local coefficients computes group cohomology.

Suppose a finite group $G$ acts freely on $S^n$, $n>1$. The quotient has fundamental group $G$ and higher homotopy groups equal to those of $S^n$. Attach one $(n+1)$-cell killing a generator of ${\pi }_n\cong \mathbb{Z}$, then attach higher cells successively to kill all remaining higher homotopy. This constructs a Eilenberg–MacLane space $K(G,1)$ with only one $(n+1)$-cell. The one-cell bound on next-degree cohomology gives dimension at most one in degree $n+1$. But if $G=C_p\times C_p$, the mod-p cohomology ring of a cyclic group and Künneth theorem give dimension $n+2$ there. This contradiction excludes that group, and also excludes such a subgroup of any group acting freely on a sphere.

For a prime $p$, a classifying Eilenberg–MacLane space for $C_p$ is the infinite lens space $S^{\infty }/C_p$, using scalar multiplication by a primitive $p$th root on the complex-coordinate sphere. Its contractible universal cover is $S^{\infty }$. The standard two-periodic free resolution has differentials $g-1$ and $1+g+\cdots +g^{p-1}$, both zero after applying cochains with trivial ${\mathbb{F}}_p$ coefficients. Thus every cohomology degree has dimension one. If $p=2$, the ring is ${\mathbb{F}}_2[t]$, $|t|=1$. If $p$ is odd, it is $\Lambda (a)\otimes {\mathbb{F}}_p[b]$, with $|a|=1$, $|b|=2$: graded commutativity gives $a^2=0$ and the periodicity class $b$ induces the degree-two isomorphisms. The Künneth theorem gives two copies of these generators for $C_p\times C_p$; the dimension in degree $d$ is $d+1$ for every $p$.

Work in the based CW homotopy category. A homotopy-invariant functor depends on a map only through its based homotopy class. In a sequence of pointed sets, exactness at the middle means that the preceding image equals the inverse image of the distinguished point; it does not imply injectivity or surjectivity at the ends. For groups this is the familiar image-equals-kernel condition. A half-exact functor preserves such middle exactness on the appropriate elementary sequence.

For a contravariant functor $F$, the relevant elementary sequence is a cofibration $A\hookrightarrow X\to X/A$. Half-exactness isexact at $F(X)$. One should not silently turn this into a short exact sequence. In particular, arbitrary group actions and pointed sets at low degrees are not interchangeable with abelian groups.

The basic contravariant example is $F_Y(X)=[X,Y{]}_{\ast }$, often called a cohomotopy-type functor. Suppose a map $f:X\to Y$ becomes null-homotopic on $A$. A chosen null-homotopy extends it over the cone attached to $A$, giving a map from the topological mapping cone to $Y$. Conversely every such extension restricts to a null-homotopic map on $A$. Since the topological mapping cone of a subcomplex inclusion models $X/A$, this proves half-exactness, rather than merely asserting it.

Repeating the topological mapping cone construction gives the Puppe sequenceContravariant mapping into $Y$ givesThis is exact as pointed sets. An H-group structure on $Y$, for example a loop space with path concatenation, provides group structures; repeated looping gives abelian group structures in the usual ranges. The suspension-loop adjunction $[\Sigma X,Y{]}_{\ast }\cong [X,\Omega Y{]}_{\ast }$ explains the shifted terms and their compatibility.

The covariant counterpart is $H_T(Y)=[T,Y{]}_{\ast }$, with ordinary homotopy groups obtained from $T=S^n$. Its natural elementary sequence is a homotopy fiber sequence $F\to E\to B$, not an arbitrary cofibration. A map $a:T\to E$ whose composite to $B$ is null-homotopic, together with that null-homotopy, determines a map into the homotopy fibre. Conversely a fibre map projects to such a null composite. This proves exactness of $[T,F{]}_{\ast }\to [T,E{]}_{\ast }\to [T,B{]}_{\ast }$ at the middle. Adding the loop terms produces the homotopy long exact sequence. For a Serre fibration, its ordinary fibre supplies the same homotopy information.

This variance distinction matters. Include one circle, generated by $a$, in $S^1\vee S^1$, with other generator $b$. The quotient is the second circle. The kernel of ${\pi }_1(S^1\vee S^1)\to {\pi }_1(S^1)$ is the normal closure of $a$, which contains $ba{b}^{-1}$. The image of the included circle is only the cyclic subgroup generated by $a$, and does not contain that reduced word. Thus covariant ${\pi }_1$ is not half-exact on this cofibration. Fibre and cofibre statements cannot be swapped just by reversing notation.

Reduced cohomology gives group-valued contravariant examples. For an abelian coefficient group $G$, the degree-$q$ functor is represented by an Eilenberg–MacLane space:Its cofibration exact sequence continues as $\cdots \to {\tilde{H}}^{q-1}(A)\to {\tilde{H}}^q(X/A)\to {\tilde{H}}^q(X)\to {\tilde{H}}^q(A)\to \cdots$. Ordinary homotopy groups are instead covariant, with decreasing group degree across a fibre connecting map. These are the two complementary representable patterns, maps out of a test object and maps into a representing object.

Brown's representability theorem makes the contravariant pattern structural. A reduced contravariant homotopy functor on based CW complexes is represented by a CW complex if it converts arbitrary wedges to products and satisfies the Mayer-Vietoris weak-pullback gluing axiom. The latter says that compatible classes on two subcomplexes extend to their union. It contains more information than the single kernel statement of half-exactness. Consequently half-exactness alone is not a representability criterion. In a representing functor these axioms are natural: a map on a wedge is a family of based maps, while compatible maps on an intersection can be adjusted by a homotopy and glued across a homotopy pushout.

A reduced generalized cohomology theory with the usual wedge axiom supplies these axioms degree by degree. Its representing spaces $E_q$ have $E_q\simeq \Omega E_{q+1}$ by suspension compatibility, assembling an Omega-spectrum. Thus generalized cohomology can be expressed as $E^q(X)=[{\Sigma }^{\infty }X,{\Sigma }^{q}E{]}_{\mathrm{st}}$, while represented homology is $E_q(X)={\pi }_q(E\wedge {\Sigma }^{\infty }X)$. The appearance of mapping into a spectrum in one case and smashing with it in the other is another form of the same variance distinction.

There is a literal duality after stabilization and a finiteness restriction. Here a finite based CW complex is identified with its suspension spectrum. For a finite spectrum $X$, let $DX=F(X,\mathbb{S})$ be its Spanier-Whitehead dual. Then $D^{2}X\simeq X$ and $[X,Y{]}_{\mathrm{st}}\cong [DY,DX{]}_{\mathrm{st}}$ for finite spectra. Duality reverses a cofibre triangle into a fibre triangle. In particular,This turns a cohomology calculation on a finite object into a homology calculation on its dual, and vice versa. It is Spanier-Whitehead duality, not a claim that the full category of ordinary spaces is self-dual.

Half-exactness is the local image–kernel property; fibres and cofibres, representability, and finite stable duality explain how that property organizes homotopy and cohomology. The distinctions between variance, pointed sets and groups, and finite stable objects and arbitrary spaces are essential to using the duality correctly.

For a based space $(X,x_0)$ and $k\ge 1$, the homotopy group ${\pi }_k(X,x_0)$ is the set of based homotopy classes of maps $(I^k,\partial I^k)\to (X,x_0)$, equivalently based maps $S^k\to X$. Concatenating along one coordinate defines its group operation; reversing that coordinate gives the inverse. The group is abelian for $k\ge 2$. The symbol ${\pi }_0(X)$ records path components and is generally only a pointed set. An Eilenberg–MacLane space $K(G,n)$ is a connected CW space with ${\pi }_n=G$ and all other positive homotopy groups zero; for $n>1$ the group $G$ must be abelian.

Projection onto coordinates givesa map to a product is exactly a pair of maps, and a based homotopy to the product is exactly a pair of based homotopies. These identifications respect concatenation. Thus $K(G,n)\times K(H,n)$ has exactly the homotopy groups required of $K(G\times H,n)$. Taking CW models and using the allowed uniqueness of Eilenberg–MacLane spaces gives

Let $p$ be prime. Take $S^{\infty }$ in complex coordinates, with $C_p$ acting by scalar multiplication by a primitive $p$th root of unity. This is a free action. Every map from a finite-dimensional sphere lands in a finite stage and becomes null-homotopic in a sufficiently higher sphere, so $S^{\infty }$ is weakly contractible; the Whitehead theorem for CW complexes makes it contractible. Therefore the infinite lens space $S^{\infty }/C_p$ has contractible universal cover, fundamental group $C_p$, and no higher homotopy: it is a model of $K(C_p,1)$.

Here is the group cohomology computation, including the ring structure. For the cyclic group generated by $g$, the standard free resolution alternates the maps $g-1$ and $N=1+g+\cdots +g^{p-1}$. Applying ${\mathrm{Hom}}_{\mathbb{Z}[C_p]}(-,{\mathbb{F}}_p)$ makes both maps zero, so each cohomology degree is one-dimensional. The period-two identity chain map represents a class $b$ in degree two, and cup multiplication by this periodicity class is an isomorphism from each degree to the degree two higher. For odd $p$, a nonzero degree-one class $a$ has $a^2=0$ by graded commutativity, while $b^j$ and $a{b}^j$ are nonzero by periodicity. For $p=2$ the model is homotopy equivalent to ${\mathbb{RP}}^{\infty }$, whose ring is the infinite version of that computed in question 2. Consequently the mod-p cohomology ring of a cyclic group isThe standard results used are the contractible-cover description of a $K(G,1)$, its identification of singular cohomology with group cohomology, and the cyclic periodic resolution with cup-periodicity just stated.

The product equivalence and the assumed Künneth theorem over the field giveThe tensor product is graded: for odd $p$, $a_1{a}_2=-a_2{a}_1$ and each $a_i^2=0$. The dimensions in degree $d$ are $d+1$: for $p=2$ count $t_1^i{t}_2^{d-i}$; for odd $p$ the generating series is $(1+t{)}^2/(1-t^2{)}^2=(1-t{)}^{-2}$.

If an $n$-dimensional CW complex has exactly one cell in dimension $n+1$, then $C^{n+1}(X;{\mathbb{F}}_p)={\mathbb{F}}_p$. Cellular cohomology is the quotient of the kernel of its outgoing coboundary by the image of the incoming one, a quotient of a subspace of this one-dimensional vector space. ThusThis one-cell bound on next-degree cohomology is valid even with infinitely many higher cells; the printed finite-per-dimension hypothesis is more than it needs.

Now a free finite-group action makes $S^n\to M=S^n/G$ a covering. For $n>1$ the sphere is simply connected, so ${\pi }_1(M)=G$, ${\pi }_i(M)=0$ for $1<i<n$, and ${\pi }_n(M)=\mathbb{Z}$. The covering also identifies every higher homotopy group with that of $S^n$. Use the homotopy-group effect of attaching higher cells: attaching $(i+1)$-cells, $i\ge 2$, preserves homotopy groups below $i$ and quotients ${\pi }_i$ by the subgroup generated by the attaching classes and their fundamental-group translates. Attach a single $(n+1)$-cell along a generator of ${\pi }_n(M)$; its translates do not require additional cells since the original generator already generates the underlying group. This kills ${\pi }_n$ without changing ${\pi }_1$ or the lower groups.

Successively in each higher degree attach cells representing generators of the remaining homotopy group. In the union every higher homotopy group vanishes, since a map and its relevant homotopy have compact domain and are handled at a finite stage. The result is a $K(G,1)$ with exactly one $(n+1)$-cell. No finiteness assumption on the number of later cells is needed for the preceding cochain bound. If desired, one can retain finitely many per degree using the Serre finiteness theorem for homotopy groups: a simply connected CW complex of finite type has finitely generated homotopy groups. At each finite stage the universal cover is simply connected and finite type, since $G$ is finite; choosing finitely many generators then preserves finite type inductively. The first stage needs just the stated ${\pi }_n(S^n)=\mathbb{Z}$.

If $G=C_p\times C_p$, the resulting $K(G,1)$ would have $\dim H^{n+1}\le 1$ by its single cell, but its already computed cohomology has dimension $n+2>1$ in that degree. This contradiction provesThis is the free sphere actions exclude elementary abelian subgroups of rank two obstruction.

Work in the homotopy category of simply connected CW complexes. This is one of the permitted assumptions on the fundamental group; it makes every coefficient system below untwisted. A rational space is a simply connected space whose homotopy groups are vector spaces over $\mathbb{Q}$. A rationalization of a topological space is a map $r:X\to X_{\mathbb{Q}}$ to such a space which inducesIts purpose includes a universal mapping property, which will be described after the construction.

An Eilenberg–MacLane space $K(A,n)$ has exactly one nonzero homotopy group, ${\pi }_n=A$, where $A$ is abelian when $n\ge 2$. The representability theorem says $[B,K(A,n)]\cong H^n(B;A)$, naturally in $B$ and $A$. More generally, a principal fibration is a homotopy pullback of a path-space fibration $PC\to C$ along a map from its base to $C$; the fibre $\Omega C$ acts on the fibre paths by concatenation. For the Postnikov tower construction take $C=K(A,n+1)$. A principal Eilenberg–MacLane fibration with fibre $K(A,n)$ is the homotopy pullback of the path fibration of $K(A,n+1)$. Equivalently, its total space is the homotopy fibre of a map $k:B\to K(A,n+1)$. It is classified by the cohomology class $[k]\in H^{n+1}(B;A)$; there is no nontrivial action of the base fundamental group on the fibre coefficients in this principal description.

The Postnikov tower theorem supplies spaces $P_{n}X$ and compatible mapswith ${\pi }_i(P_{n}X)={\pi }_i(X)$ for $i\le n$ and zero for $i>n$. For a simply connected $X$, the fibre of $P_{n}X\to P_{n-1}X$ is $K({\pi }_{n}X,n)$, and the fibration is principal. Its Postnikov invariant isrepresented by $P_{n-1}X\to K({\pi }_{n}X,n+1)$, whose homotopy fibre is $P_{n}X$. Thus the homotopy groups alone are not the tower data: the classes $k_n$ specify how each new group is attached. For instance a zero class gives a product with $K({\pi }_{n}X,n)$.

State also the standard rational comparison results used in constructing the tower: $K(A,n)\to K(A\otimes \mathbb{Q},n)$ is a rational homology equivalence; a map between simply connected spaces is a rational homology equivalence if and only if it induces isomorphisms on all rational homotopy groups; and a rational homology equivalence induces isomorphisms in cohomology with any constant rational vector-space coefficients. The last assertion uses the universal coefficient theorem over the field $\mathbb{Q}$. These results are used here without proof, as allowed for this essay.

Construct rational stages inductively, with $P_1^{\mathbb{Q}}=\ast$. Suppose $r_{n-1}:P_{n-1}X\to P_{n-1}^{\mathbb{Q}}$ has already been constructed and is a rational homology equivalence. Put $V_n={\pi }_{n}X\otimes \mathbb{Q}$ and change coefficients in $k_n$ to obtain ${\overline{k}}_n\in H^{n+1}(P_{n-1}X;V_n)$. The comparison result gives an isomorphismLet $k_n^{\mathbb{Q}}$ be its unique inverse image and defineEquality of the pulled-back classes gives a homotopy-commuting square of classifying maps, and hence a map $r_n:P_{n}X\to P_n^{\mathbb{Q}}$ over $r_{n-1}$. On the fibre it is the map induced by ${\pi }_{n}X\to V_n$. The homotopy exact sequences therefore show that $r_n$ tensors all homotopy groups through degree $n$ with $\mathbb{Q}$, while the higher groups of both stages vanish. The rational comparison theorem makes $r_n$ a rational homology equivalence, allowing the induction to continue.

Take the homotopy inverse limit of the rational tower:The convergence theorem for a Postnikov tower gives $X\simeq {\mathrm{holim}}_n{P}_{n}X$ and the compatible stage maps induce $r:X\to X_{\mathbb{Q}}$. To see what convergence says here, fix a homotopy degree $i$. From stage $n\ge i$ onwards the group and its transition maps are constant; the inverse-limit group is ${\pi }_{i}X\otimes \mathbb{Q}$ and the associated derived-limit obstruction is zero. Thus this construction has exactly the claimed rational homotopy groups. Taking a homotopy limit, rather than an arbitrary set-theoretic inverse limit, is part of the construction.

The rational localization theorem says that this map has the universal propertyIn other words every map from $X$ to a rational space factors through $r$, and the factorization is unique up to homotopy. This concerns homotopy classes, not strict uniqueness of point-set maps. The theorem can be formulated as localization of the simply connected homotopy category at rational homotopy equivalences; rational spaces are its local objects. It also implies uniqueness of rationalization of a topological space: apply the property to two candidate rational spaces to obtain inverse equivalences under $X$. In particular the construction forgets torsion in homotopy while retaining precisely the information needed for maps to rational targets.

The same theory extends to nilpotent spaces, where the fundamental group is nilpotent and its action on higher homotopy groups is nilpotent. That version requires localization of the fundamental group and compatible treatment of its coefficient action. The simply connected assumption chosen here avoids those additional issues and supplies the full construction and mapping property requested.

A Serre fibration has the homotopy lifting property for disks: for every $k\ge 0$, a map $D^k\to E$ and a homotopy $D^k\times I\to B$ starting at its composite with $p:E\to B$ admit a compatible lift $D^k\times I\to E$. Equivalently, it has that lifting property for CW complexes. Relative lifting for CW pairs follows by attaching cells.

Let $i:F\hookrightarrow E$ be the inclusion of the chosen fiber. The long exact sequence of homotopy groups of a fibration isIts low-dimensional end isThe maps $i_{\ast }$ and $p_{\ast }$ are induced by inclusion and projection on based maps; at degree zero they send a path component to its containing or image component. The last part is an exact sequence of pointed sets, not in general a sequence of groups.

For $\partial :{\pi }_q(B)\to {\pi }_{q-1}(F)$, represent a class by a based cube $a:I^{q-1}\times I\to B$, constant at $b_0$ on its boundary. Lift it from the constant map $e_0$ on the bottom and side faces, using relative disk lifting. Its top face lands in $F$ and is constant on that face's boundary; its class is $\partial [a]$. The choice of lifts does not change the resulting class. For $q=1$, lift a based loop starting at $e_0$ and take the path component of its endpoint in $F$. This fixes the boundary-map convention and defines every map in the displayed sequence.

For the splitting assertion, first take $n\ge 1$. We use the cohomological Serre spectral sequence with coefficient group $G$:Since $B$ is simply connected, these coefficient systems are constant. The Eilenberg–MacLane space $F=K(G,n)$ has $H^t(F;G)=0$ for $0<t<n$, andFor $n=1$ this uses $H_1(F)=G$, with $G$ abelian; for $n>1$ it follows from the Hurewicz theorem and the universal coefficient theorem for cohomology. Let $u$ correspond to ${\mathrm{id}}_G$, the universal cohomology class of an Eilenberg–MacLane space.

In bidegree $(0,n)$ there are no incoming differentials. For $2\le r\le n$, an outgoing differential lands in a vanishing fiber-cohomology row. The only remaining possible differential isThus $u$ survives. The edge map, which is restriction to $F$, supplies a classBy representability of cohomology by Eilenberg–MacLane spaces, choose a based map $q:E\to K(G,n)$ representing $c$. On $F$, $q$ induces the identity on ${\pi }_n=G$ and is an isomorphism on every homotopy group, since the other positive groups vanish.

Consider $h=(p,q):E\to B\times K(G,n)$. This is a map of fibrations over $B$, whose map on the fiber is the weak equivalence just obtained. The map on the base is the identity. Comparing the two instances of the long exact sequence of homotopy groups of a fibration and applying exactness, or the Five lemma in the group-valued range, shows that $h$ induces all homotopy isomorphisms. The spaces are path-connected because the base and fiber are. Hence the splitting of a simply connected Eilenberg–MacLane fibration givesThe crucial use of $H^{n+1}(B;G)=0$ is to lift the fiber's identity class; existence of a section alone was not assumed to prove a product splitting.

If $K(G,0)$ is allowed, its components have trivial positive homotopy groups. The simply connected base has no monodromy on this discrete set. Obstruction theory on the CW base gives a section in each fiber component, since all positive-dimensional fiber obstructions vanish and the component system is constant. Together these sections give $B\times G\to E$, a weak equivalence on each component. Thus the same conclusion also covers that interpretation.

Let $u_n$ be the rationalized universal class of degree $n$. The rational cohomology of an integral Eilenberg–MacLane space isHere $\Lambda$ is an exterior algebra. For $n=1$, $K(\mathbb{Z},1)\simeq S^1$. The standard path-loop spectral-sequence calculation supplies the induction: inthe total space is contractible and the fundamental fiber class transgresses to $u_n$. An odd exterior fiber generator gives an even polynomial base generator. An even polynomial fiber generator gives an odd exterior base generator; the differential on its $k$th power has coefficient $k$, which is invertible over $\mathbb{Q}$. The multiplicative spectral sequence then has no remaining positive-degree classes in the total space. This is the rational transgression calculation for Eilenberg–MacLane spaces; it includes the absence of additional base generators.

For $m=2n+1\ge 3$, choose $S^m\to K(\mathbb{Z},m)$ representing the integral fundamental class. The ring calculation shows that this map is a rational homology equivalence: both spaces have rational cohomology only in degrees zero and $m$. The rational Whitehead theorem for simply connected spaces identifies their rational homotopy groups. Since the target has only ${\pi }_m=\mathbb{Z}$, the rational homotopy groups of a sphere in odd dimension areFor $n=0$, this follows directly from ${\pi }_1(S^1)=\mathbb{Z}$ and the contractible universal cover of the circle, which makes every higher homotopy group zero.

Let $M={\mathbb{CP}}^2\#{\mathbb{CP}}^2$, with the standard complex orientations. It is simply connected by the Seifert-van Kampen theorem applied to the punctured summands. Classes $a,b\in H^2(M;\mathbb{Q})$ can be chosen from the two summands. Their cross product vanishes, while their squares equal the oriented top class:Thus the cohomology ring of the connected sum of two complex projective planes isThe two relations also kill all cubic monomials, so its dimensions are $1,2,1$ in degrees $0,2,4$ and zero otherwise.

We use the Sullivan minimal model dictionary: for a simply connected finite-type space, the dual of its degree-$i$ generator space is ${\pi }_i(M)\otimes \mathbb{Q}$. The following free graded-commutative differential algebra is the Sullivan model of the connected sum of two complex projective planes:It is minimal because all differentials of generators are decomposable.

To verify that no further generators are required, observe that $ab,a^2-b^2$ is a regular sequence in $\mathbb{Q}[a,b]$. The first polynomial is a nonzerodivisor. If $(a^2-b^2)h$ is divisible by $ab$, restricting to each coordinate axis forces $h$ to vanish on both axes, hence to be divisible by $ab$. The second polynomial is therefore a nonzerodivisor modulo the first. The Koszul complex of this regular sequence is exactly $\mathcal{M}$, so its cohomology is the quotient ring above, with no additional odd cohomology.

For completeness, choose rational polynomial forms representing $a,b$ on $M$. Their product and the difference of their squares are exact; choose degree-three primitives for them. Sending $r,s$ to those primitives defines a differential-algebra map from $\mathcal{M}$ to the rational polynomial forms on $M$. It induces the specified cohomology-ring isomorphism and hence is a quasi-isomorphism. This verifies the model directly, rather than assuming that a cohomology presentation alone automatically determines all rational homotopy.

The model has exactly two degree-two and two degree-three generators. Consequently the rational homotopy groups of the connected sum of two complex projective planes are

Choose the Hopf map $\eta$ and orientations so its Hopf invariant is one. The cellular cohomology of $Y_d$ is $\mathbb{Z}$ in degrees $0,2,4$ and zero otherwise. Let $x,z$ be generators in degrees two and four. Precomposing $\eta$ with a degree-$d$ self-map of $S^3$ represents $d\eta$. The induced map of mapping cones is the identity on the two-cell and has degree $d$ on the four-cell. Since the Hopf mapping cone has cup square equal to its top generator, naturality of the cup product gives $x^2=dz$. Equivalently, precomposition scales the Hopf invariant by degree. HenceThe coefficient is $d$, not $d^2$: multiplication in ${\pi }_3(S^2)$ is being used, rather than postcomposition by a degree-$d$ map of the target sphere.

For $X=X_{d,f}$, the only nonzero cellular boundary is $C_4\to C_3$, multiplication by $f$. Thus its cellular cohomology givesLet $x$ again generate $H^2$, and let $z$ be the class of the four-cell cochain. Collapsing the three-sphere gives a map $X\to Y_d$ which pulls the degree-two generator back to $x$ and the degree-four generator back to $z$. Hence $x^2=dz$, now interpreted modulo $f$. Every other product of positive-degree classes vanishes by dimension. This is the cohomology ring of a Hopf attachment with a sphere summand. More explicitly, if $f\ne 0$,If $f=0$, there is additionally a generator $y$ of degree three, with $xy=y^2=yz=0$. The displayed description of the groups uses $\mathbb{Z}/0\mathbb{Z}=\mathbb{Z}$, so the zero case is included. In particular, a nonzero $f$ kills the degree-three cohomology but can leave torsion in degree four.

There are no one-cells, and attaching cells of dimension at least three does not change ${\pi }_1$. Thus $X$ is simply connected. The Hurewicz theorem in degree two gives

To compute ${\pi }_3$, choose a map $g:X\to {\mathbb{CP}}^{\infty }=K(\mathbb{Z},2)$ representing $x$. This Eilenberg–MacLane space can be realized by the classifying space of $S^1$. On $S^2$ the map is the standard inclusion; on $S^3$ it is constant, and it extends over the four-cell since ${\pi }_3(K(\mathbb{Z},2))=0$. Its homotopy fibre $T$ is the pullback of the universal circle bundle $S^{\infty }\to {\mathbb{CP}}^{\infty }$. The map on ${\pi }_2$ is an isomorphism, so the long exact sequence of homotopy groups of a fibration gives ${\pi }_1(T)={\pi }_2(T)=0$ and ${\pi }_3(T)\cong {\pi }_3(X)$. Therefore the Hurewicz theorem identifies the latter with $H_3(T;\mathbb{Z})$.

Over $W=S^2\vee S^3$, the circle-bundle total space isThe first summand is the Hopf fibration total space over $S^2$, and the second is the trivial bundle over $S^3$. Their intersection is the fibre over the wedge point. The Mayer–Vietoris sequence gives $H_3(P)=\mathbb{Z}A\oplus \mathbb{Z}B$, with generators from the two three-spheres. These are the lifts of ${\eta }_1$ and $i_2$. The circle bundle over $W$ also has a 2-connected total space, so lifting and the Hurewicz theorem identify $d{\eta }_1+f{i}_2$ with $dA+fB$.

Over the attached four-disc the bundle is trivial. The resulting relative pair has the homology of $(S^1\times D^4,S^1\times S^3)$, so $H_4(T,P)=\mathbb{Z}$ and $H_3(T,P)=0$. Its boundary map sends a generator to the lift of the attaching map, namely $dA+fB$. The long exact sequence in relative homology now givesApplying Smith normal form to this one relation proves the third homotopy group of a Hopf attachment with a sphere summand:Since $d>0$, the greatest common divisor is positive even when $f=0$. This derivation uses the required map to an Eilenberg–MacLane space and determines the extension, rather than merely the orders of its pieces.

Let $A[n]$ be the nonnegative chain complex having $A$ in degree $n$, zero in every other degree, and zero differential. The simplicial Eilenberg–MacLane space isIts underlying simplicial set is a Kan complex, with ${\pi }_{n}K(A,n)\cong A$ and all other positive homotopy groups zero.

Products of Eilenberg–MacLane spaces satisfywhere the last isomorphism uses the Chinese remainder theorem. This space is $2$-connected, so the Hurewicz theorem identifiesThe assumed surjection on $H_3$ is an isomorphism because the group is finite. Hence $f$ is an isomorphism on ${\pi }_3$; all other homotopy groups of the source and target vanish. Thus $f$ is a weak homotopy equivalence, and the Whitehead theorem for Kan complexes makes it a homotopy equivalence.

For a CW complex $X$, abelian $G$ and $n\ge 1$, homotopy classes of maps to the Eilenberg–MacLane space correspond to cohomology classes by pullback of the universal cohomology class of an Eilenberg–MacLane space. Pointed maps give the reduced version. This converts a cohomology class into an actual map realizing its effect on homotopy.