We present a short proof of the Künneth theorem due to Heller. We also include a short proof of the Eilenberg-Zilber theorem, which is needed for the topological examples, using acyclic models following Boardman.

1. Prerequisites

Let \(R\) be a principal ideal domain, and let all modules be left \(R\)-modules.

For \(R\)-modules \(M\) and \(N\), the tensor product \(M\otimes_R N\) is an \(R\)-module together with a bilinear map \(\tau\colon M\times N\to M\otimes_R N\), written \(\tau(m,n)=m\otimes n\), such that for every \(R\)-module \(P\) and every bilinear map \(\beta\colon M\times N\to P\), there is a unique linear map \(f\colon M\otimes_R N\to P\) making the following diagram commute.

For fixed \(R\)-modules \(N\) and \(G\), there is a natural isomorphism

\[ \Hom_R(M\otimes_R N,G)\cong \Hom_R(M,\Hom_R(N,G)). \]

Given \(\varphi\in \Hom_R(M\otimes_R N,G)\), define

\[ \Phi(\varphi)(m)(n)=\varphi(m\otimes n). \]

Given \(\psi\in \Hom_R(M,\Hom_R(N,G))\), define

\[ \Psi(\psi)(m\otimes n)=\psi(m)(n) \]

on simple tensors, and extend linearly to \(M\otimes_R N\). It is an easy check that both maps are well defined, \(R\)-linear, natural, and inverse to each other.

For every \(R\)-module \(N\), the functor \(-\otimes_R N\) is right exact. For every \(R\)-module \(N\), the functor \(\Hom_R(N,-)\) is left exact.

By , the functor \(-\otimes_R N\) is left adjoint to \(\Hom_R(N,-)\), so it preserves colimits, in particular cokernels. Since it is additive, it is right exact. Dually, \(\Hom_R(N,-)\) is a right adjoint, so it preserves limits, in particular kernels. Since it is additive, it is left exact.

A module \(P\) is projective if whenever \(\pi \colon M \twoheadrightarrow N\) is surjective and \(f \colon P \to N\) is a homomorphism, there exists \(\widetilde f \colon P \to M\) such that \(\pi \circ \widetilde f = f\).

A module \(M\) is flat if the functor \(- \otimes_R M\) is exact.

Every free module is projective, and every projective module is flat.

A free module has a basis, so maps out of it are determined by the images of basis elements, and those images can be lifted across a surjection. Thus free modules are projective. The module \(R\) is flat because \(- \otimes_R R \cong -\), direct sums of flat modules are flat, and projective modules are direct summands of free modules. A direct summand of a flat module is flat.

Every submodule of a free \(R\)-module is free.

This is the standard fact that submodules of free modules over a PID are free.

Every \(R\)-module admits a free resolution of length \(1\). In particular, every \(R\)-module has projective dimension at most \(1\).

Choose a surjection \(F_0 \twoheadrightarrow M\) with \(F_0\) free, and set \(F_1=\ker(F_0\to M)\). By , the module \(F_1\) is free, so

\[ 0 \to F_1 \to F_0 \to M \to 0 \]

is a free resolution of length \(1\).

This is the reason only \(\Tor_1\) and \(\Ext^1\) appear later.

2. Resolutions and Derived Functors

Let \(P_\bullet \to M\) be a projective resolution and let \(G_\bullet \to N\) be an exact complex with \(H_0(G_\bullet)\cong N\) and \(H_i(G_\bullet)=0\) for \(i>0\). Every map \(u \colon M \to N\) extends to a chain map \(P_\bullet \to G_\bullet\), and any two such extensions are chain homotopic.

Lift \(u \circ (P_0 \to M)\) through \(G_0 \twoheadrightarrow N\) using projectivity of \(P_0\). If \(P_n \to G_n\) has already been constructed, its composite with the differential lands in \(\ker(G_n \to G_{n-1})=\operatorname{im}(G_{n+1}\to G_n)\), so projectivity of \(P_{n+1}\) produces the next lift. Induction gives the chain map.

For uniqueness, apply the same lifting argument to the difference of two lifts. Exactness of \(G_\bullet\) produces maps \(h_n \colon P_n \to G_{n+1}\) with

\[ \alpha-\beta = dh + hd. \]

Any two projective resolutions of the same module are chain homotopy equivalent.

Apply twice to the identity map of the module.

If \(F\) is an additive right exact covariant functor and \(P_\bullet \to M\) is a projective resolution, define

\[ L_iF(M)=H_i(FP_\bullet). \]

If \(T\) is an additive left exact covariant functor and \(M \to I^\bullet\) is an injective resolution, define

\[ R^iT(M)=H^i(TI^\bullet). \]

These functors are characterized by the formal properties below. In other words, \(L_\ast F\) is the universal homological \(\delta\)-functor extending \(F\). See Weibel, Theorem 2.4.7.

The left derived functors are independent of the chosen projective resolution. They satisfy the following standard properties:

1. \(L_0F \cong F\).
2. If \(P\) is projective, then \(L_iF(P)=0\) for every \(i>0\).
3. Every short exact sequence \(0 \to A \xrightarrow{\alpha} B \xrightarrow{\beta} C \to 0\) induces a natural long exact sequence

1. The connecting homomorphisms are natural. If

is a morphism of short exact sequences, then the induced square

commutes.

Define

\[ \Tor_i^R(M,N)=L_i(-\otimes_R N)(M), \qquad \Ext_R^i(M,N)=R^i\Hom_R(M,-)(N). \]

By , only \(i=0,1\) occur over a PID.

3. Heller’s Construction

The key step in Heller’s proof Heller is the exact sequence argument assembled in the next lemmas and then closed by the patching lemma.

If \((C_\bullet,d_C)\) and \((D_\bullet,d_D)\) are chain complexes, define

\[ (C\otimes_R D)_n=\bigoplus_{p+q=n} C_p\otimes_R D_q \]

with differential

\[ d(c\otimes d)=d_C(c)\otimes d + (-1)^p c\otimes d_D(d) \qquad (c\in C_p). \]

For a complex \(C_\bullet\), the graded groups \(Z(C)\), \(B(C)\), and \(H(C)\) are viewed as complexes with zero differential.

If each \(C_n\) is free, then each \(Z_n(C)\) and \(B_n(C)\) is free.

Both are submodules of free modules, so applies.

If \(C_\bullet\) is degreewise free, then there is a short exact sequence of chain complexes

\[ 0 \to Z(C) \xrightarrow{\iota} C \xrightarrow{d} B(C)[1] \to 0, \]

where \(B(C)[1]_n=B_{n-1}(C)\). This sequence splits in each degree.

Exactness is immediate. By , each \(B_{n-1}(C)\) is free, so the surjection \(d_n \colon C_n \twoheadrightarrow B_{n-1}(C)\) has a section.

If \(F=\bigoplus_p F_p\) is a graded free module with zero differential, then

\[ H_n(F\otimes_R D)\cong \bigoplus_{p+q=n} F_p\otimes_R H_q(D) \]

naturally in \(D\).

The differential acts only on the \(D\)-factor, and each \(F_p\) is flat by .

Assume \(C_\bullet\) is degreewise free. Tensor the short exact sequence in with \(D_\bullet\). Under the identifications

\[ H_n(B(C)[1]\otimes_R D)\cong \bigl(B(C)\otimes_R H(D)\bigr)_{n-1} \]

and

\[ H_{n-1}(Z(C)\otimes_R D)\cong \bigl(Z(C)\otimes_R H(D)\bigr)_{n-1}, \]

the connecting map

\[ H_n(B(C)[1]\otimes_R D)\to H_{n-1}(Z(C)\otimes_R D) \]

is induced by the inclusion \(j \colon B(C)\hookrightarrow Z(C)\).

Choose a degreewise section \(q \colon B(C)[1]\to C\) of the differential. Then \(d_C q=j\) after shifting degrees. The standard description of the connecting homomorphism together with gives the claim.

Suppose that for each \(n\) we have an exact sequence

\[ A_n \xrightarrow{f_n} B_n \xrightarrow{u_n} X_n \xrightarrow{v_n} A_{n-1} \]

and a short exact sequence

\[ 0 \to T_n \xrightarrow{\lambda_n} A_n \xrightarrow{f_n} B_n \xrightarrow{\rho_n} S_n \to 0. \]

Then there are unique maps \(I_n \colon S_n \to X_n\) and \(K_n \colon X_n \to T_{n-1}\) fitting into a short exact sequence

\[ 0 \to S_n \xrightarrow{I_n} X_n \xrightarrow{K_n} T_{n-1} \to 0. \]

Because \(u_n f_n=0\), the map \(u_n\) factors uniquely through the cokernel \(S_n\). Write \(u_n=I_n\rho_n\). Because \(f_{n-1}v_n=0\), the image of \(v_n\) lies in \(\ker f_{n-1}=\operatorname{im}\lambda_{n-1}\), so \(v_n\) factors uniquely through \(T_{n-1}\). Write \(v_n=\lambda_{n-1}K_n\). These factorizations give the commutative diagram

We now verify exactness.

If \(I_n(s)=0\), choose \(b\in B_n\) with \(\rho_n(b)=s\). Then \(0=I_n\rho_n(b)=u_n(b)\), so exactness gives \(b=f_n(a)\) for some \(a\in A_n\). Hence \(s=\rho_n(b)=0\). So \(I_n\) is injective.

For \(s\in S_n\), choose \(b\in B_n\) with \(\rho_n(b)=s\). Then \(\lambda_{n-1}K_nI_n(s)=v_nI_n\rho_n(b)=v_nu_n(b)=0\), so \(K_nI_n(s)=0\). Thus \(\operatorname{im}I_n\subseteq\ker K_n\). Conversely, if \(K_n(x)=0\), then \(v_n(x)=\lambda_{n-1}K_n(x)=0\), so exactness gives \(x=u_n(b)\) for some \(b\in B_n\). Then \(x=I_n\rho_n(b)\). Hence \(\ker K_n\subseteq\operatorname{im}I_n\).

Finally, let \(t\in T_{n-1}\). Since \(\lambda_{n-1}(t)\in\ker f_{n-1}\), exactness of \(X_n\xrightarrow{v_n}A_{n-1}\xrightarrow{f_{n-1}}B_{n-1}\) gives \(x\in X_n\) with \(v_n(x)=\lambda_{n-1}(t)\). Then \(\lambda_{n-1}K_n(x)=v_n(x)=\lambda_{n-1}(t)\), so \(K_n(x)=t\). Thus \(K_n\) is surjective. The sequence is exact.

4. Künneth over a PID

Let \(C_\bullet\) and \(D_\bullet\) be bounded-below chain complexes of \(R\)-modules, and assume each \(C_n\) is free. Then for each \(n\) there is a natural short exact sequence

\[ 0 \to \bigoplus_{p+q=n} H_p(C)\otimes_R H_q(D) \xrightarrow{I_n} H_n(C\otimes_R D) \xrightarrow{K_n} \bigoplus_{p+q=n-1}\Tor_1^R(H_p(C),H_q(D)) \to 0. \]

Set

\[ A_n=\bigoplus_{p+q=n} B_p(C)\otimes_R H_q(D), \qquad B_n=\bigoplus_{p+q=n} Z_p(C)\otimes_R H_q(D), \]

\[ X_n=H_n(C\otimes_R D), \qquad S_n=\bigoplus_{p+q=n} H_p(C)\otimes_R H_q(D), \]

and

\[ T_n=\bigoplus_{p+q=n}\Tor_1^R(H_p(C),H_q(D)). \]

Heller’s exact triangle in degree \(n\) is the fragment

\[ \begin{array}{ccc} A_n & \xrightarrow{j_n} & B_n
& & \downarrow u_n
A_{n-1} & \xleftarrow{v_n} & X_n \end{array} \]

and the proof consists of patching this triangle to the short exact sequence

\[ 0 \to T_n \to A_n \xrightarrow{j_n} B_n \to S_n \to 0. \]

Tensor with \(D_\bullet\). Using and , the long exact sequence in homology becomes

\[ A_n \xrightarrow{j_n} B_n \xrightarrow{u_n} X_n \xrightarrow{v_n} A_{n-1}. \]

For each \(p\), the short exact sequence

\[ 0 \to B_p(C) \xrightarrow{j} Z_p(C) \to H_p(C) \to 0 \]

is a free resolution of \(H_p(C)\) of length \(1\). After tensoring with \(H_q(D)\) we get

\[ 0 \to \Tor_1^R(H_p(C),H_q(D)) \to B_p(C)\otimes_R H_q(D) \xrightarrow{j\otimes 1} Z_p(C)\otimes_R H_q(D) \to H_p(C)\otimes_R H_q(D) \to 0. \]

Summing over \(p+q=n\) yields

\[ 0 \to T_n \to A_n \xrightarrow{j_n} B_n \to S_n \to 0. \]

Now applies to these two exact sequences and produces

\[ 0 \to S_n \xrightarrow{I_n} X_n \xrightarrow{K_n} T_{n-1} \to 0, \]

which is exactly the stated Künneth short exact sequence. The left-hand map is induced by \([z]\otimes[d]\mapsto[z\otimes d]\).

5. Dual Form and Universal Coefficients

If \((C_\bullet,d_C)\) and \((D_\bullet,d_D)\) are chain complexes, define

\[ \Hom^n(C,D)=\prod_p \Hom_R(C_p,D_{p+n}) \]

with differential

\[ (df)_p=d_Df_p-(-1)^n f_{p-1}d_C. \]

Let \(C_\bullet\) and \(D_\bullet\) be bounded-below chain complexes of \(R\)-modules, with each \(C_n\) free. Then for each \(n\) there is a natural short exact sequence

\[ 0 \to \prod_p \Ext_R^1(H_p(C),H_{p+n-1}(D)) \to H^n\Hom^\bullet(C,D) \to \prod_p \Hom_R(H_p(C),H_{p+n}(D)) \to 0. \]

This is the formal dual of : apply \(\Hom^\bullet(-,D)\) to \(0 \to B(C) \to Z(C) \to H(C) \to 0\) and compare the resulting cohomology sequence with the low-degree \(\Ext\) sequence coming from a free resolution of each \(H_p(C)\).

Let \(C_\bullet\) be a bounded-below chain complex of free abelian groups and let \(G\) be an abelian group concentrated in degree \(0\). Then for each \(n\) there is a natural short exact sequence

\[ 0 \to H_n(C)\otimes G \to H_n(C\otimes G) \to \Tor_1^{\mathbb Z}(H_{n-1}(C),G) \to 0. \]

Apply with \(R=\mathbb Z\) and \(D_\bullet=G[0]\).

Let \(C_\bullet\) be a bounded-below chain complex of free abelian groups and let \(G\) be an abelian group. Then for each \(n\) there is a natural short exact sequence

\[ 0 \to \Ext_{\mathbb Z}^1(H_{n-1}(C),G) \to H^n\Hom^\bullet(C,G[0]) \to \Hom(H_n(C),G) \to 0. \]

Apply with \(R=\mathbb Z\) and \(D_\bullet=G[0]\).

6. Eilenberg-Zilber

We follow Boardman’s note Boardman, but phrase the argument in the standard language of categories with models.

A category with models is a category \(\mathcal C\) together with a chosen set \(\mathcal M\) of objects.

Let \((\mathcal C,\mathcal M)\) be a category with models. A functor \(F\colon \mathcal C\to \mathrm{Ab}\) is \(\mathcal M\)-free if it is a direct sum of functors of the form \(\mathbb Z\,\mathcal C(M,-)\) with \(M\in\mathcal M\).

Let \(G^\prime \to G \to G^{\prime\prime}\) be a composable pair of natural transformations with zero composite. Write \(K\subseteq G\) for the objectwise kernel of \(G\to G^{\prime\prime}\). We say that the sequence is \(\mathcal M\)-exact if the induced map \((G^\prime)(M)\to K(M)\) is surjective for every \(M\in\mathcal M\). Equivalently, \((G^\prime)(M)\to G(M)\to G^{\prime\prime}(M)\) is exact for every model object \(M\).

Let \((\mathcal C,\mathcal M)\) be a category with models. If \(F\) is \(\mathcal M\)-free and \(G^\prime \to G\) is \(\mathcal M\)-epimorphic, then every natural transformation \(f\colon F\to G\) lifts to a natural transformation \(\widetilde f\colon F\to G^\prime\).

It is enough to treat one summand \(F=\mathbb Z\,\mathcal C(M,-)\). A natural transformation \(f\colon \mathbb Z\,\mathcal C(M,-)\to G\) is determined by the value of \(f_M(1_M)\in G(M)\). Since \((G^\prime)(M)\to G(M)\) is surjective, choose \(c_M\in (G^\prime)(M)\) mapping to \(f_M(1_M)\). For \(\phi\colon M\to X\), define \(\widetilde f_X(\phi)=G^\prime(\phi)(c_M)\), and extend linearly. This is natural and lifts \(f\).

Let \(\theta\colon F\to G\) be a natural transformation of functors \(\mathcal C\to \mathrm{Ab}\). Let \(F_\bullet\) and \(G_\bullet\) be augmented chain-complex-valued functors with augmentations \(F_0\to F\) and \(G_0\to G\). Assume that each \(F_n\) is \(\mathcal M\)-free and that

\[ \cdots \to G_2 \to G_1 \to G_0 \to G \to 0 \]

is \(\mathcal M\)-exact. Then there is a chain map \(F_\bullet\to G_\bullet\) covering \(\theta\), and its chain homotopy class is unique.

Construct the map degree by degree. Suppose \(\alpha_i\colon F_i\to G_i\) has been defined for \(i<n\) and satisfies \(\partial\alpha_i=\alpha_{i-1}\partial\). Then \(\alpha_{n-1}\partial\colon F_n\to G_{n-1}\) lands in \(\ker(\partial\colon G_{n-1}\to G_{n-2})\). Since \(F_n\) is \(\mathcal M\)-free and \(G_n\to \ker(\partial)\) is \(\mathcal M\)-epimorphic by \(\mathcal M\)-exactness, produces \(\alpha_n\colon F_n\to G_n\) with \(\partial\alpha_n=\alpha_{n-1}\partial\).

The homotopy statement is proved in the same way. If \(\alpha,\beta\colon F_\bullet\to G_\bullet\) cover the same \(\theta\), then one constructs a chain homotopy \(h_n\colon F_n\to G_{n+1}\) inductively from the fact that \(\alpha_n-\beta_n-h_{n-1}\partial\) lands in \(\ker(\partial\colon G_n\to G_{n-1})\), and again applies .

Let \(\mathcal C=\mathrm{Top}^2\), and let \(\mathcal M={(\Delta^p,\Delta^q)\mid p,q\ge 0}\). For each \(n\ge 0\), the functors

\[ E_n(X,Y)=C_n(X\times Y), \qquad G_n(X,Y)=\bigoplus_{p+q=n} C_p(X)\otimes C_q(Y) \]

are \(\mathcal M\)-free.

The group \(E_n(X,Y)\) is freely generated by singular simplices \(\sigma\colon \Delta^n\to X\times Y\), so \(E_n\cong \mathbb Z\,\mathcal C((\Delta^n,\Delta^n),-)\). Likewise, \(C_p(X)\otimes C_q(Y)\) is free on pairs of singular simplices \((\sigma,\tau)\), so \(C_p(X)\otimes C_q(Y)\cong \mathbb Z\,\mathcal C((\Delta^p,\Delta^q),-)\). Summing over \(p+q=n\) gives the claim for \(G_n\).

With \(\mathcal C\) and \(\mathcal M\) as above, the augmented sequences

\[ \cdots \to E_2 \to E_1 \to E_0 \to H_0(X\times Y) \to 0 \]

and

\[ \cdots \to G_2 \to G_1 \to G_0 \to H_0(X)\otimes H_0(Y) \to 0 \]

are \(\mathcal M\)-exact.

Evaluate at a model \((\Delta^p,\Delta^q)\). Since \(\Delta^p\times\Delta^q\) is contractible, the augmented singular complex \(C_\bullet(\Delta^p\times\Delta^q)\to H_0(\Delta^p\times\Delta^q)\to 0\) is exact. Also \(\Delta^p\) and \(\Delta^q\) are contractible, so each augmented complex \(C_\bullet(\Delta^p)\to H_0(\Delta^p)\to 0\) and \(C_\bullet(\Delta^q)\to H_0(\Delta^q)\to 0\) is chain homotopy equivalent to \(\mathbb Z\) in degree \(0\). Their tensor product is therefore chain homotopy equivalent to \(\mathbb Z\otimes\mathbb Z\cong\mathbb Z\), which gives exactness of \(G_\bullet(\Delta^p,\Delta^q)\to H_0(\Delta^p)\otimes H_0(\Delta^q)\to 0\).

For spaces \(X\) and \(Y\), the singular chain complexes \(C_\bullet(X\times Y)\) and \(C_\bullet(X)\otimes C_\bullet(Y)\) are naturally chain homotopy equivalent.

Let \(E_\bullet(X,Y)=C_\bullet(X\times Y)\) and \(G_\bullet(X,Y)=C_\bullet(X)\otimes C_\bullet(Y)\). By , both functors are degreewise \(\mathcal M\)-free. By , both augmented sequences are \(\mathcal M\)-exact.

The degree-zero cross product gives the usual natural isomorphism \[ \theta\colon H_0(X)\otimes H_0(Y)\xrightarrow{\ \cong\ } H_0(X\times Y). \] Applying to \(\theta\) gives a chain map \(\alpha\colon G_\bullet\to E_\bullet\), and applying it to \(\theta^{-1}\) gives a chain map \(\beta\colon E_\bullet\to G_\bullet\).

The composites \(\beta\alpha\) and \(\alpha\beta\) cover the identity on degree zero. By , each composite has the unique chain homotopy class with that property. The identity chain maps have the same property, so \(\beta\alpha\) is chain homotopic to the identity on \(G_\bullet\), and \(\alpha\beta\) is chain homotopic to the identity on \(E_\bullet\). Thus \(\alpha\) and \(\beta\) are natural chain homotopy inverses.

7. Passage to Topology

Each singular chain group \(C_n(X)\) is free abelian on singular \(n\)-simplices, so the algebraic theorems above apply to singular chains.

For spaces \(X\) and \(Y\), and every \(n\ge 0\), there is a natural short exact sequence

\[ 0 \to \bigoplus_{p+q=n} H_p(X;\mathbb Z)\otimes H_q(Y;\mathbb Z) \to H_n(X\times Y;\mathbb Z) \to \bigoplus_{p+q=n-1}\Tor_1^{\mathbb Z}(H_p(X;\mathbb Z),H_q(Y;\mathbb Z)) \to 0. \]

Apply to singular chains and use .

For every space \(X\), every abelian group \(G\), and every \(n\ge 0\), there is a natural short exact sequence

\[ 0 \to H_n(X;\mathbb Z)\otimes G \to H_n(X;G) \to \Tor_1^{\mathbb Z}(H_{n-1}(X;\mathbb Z),G) \to 0. \]

Apply to \(C_\bullet(X)\).

For every space \(X\), every abelian group \(G\), and every \(n\ge 0\), there is a natural short exact sequence

\[ 0 \to \Ext_{\mathbb Z}^1(H_{n-1}(X;\mathbb Z),G) \to H^n(X;G) \to \Hom(H_n(X;\mathbb Z),G) \to 0. \]

Apply to \(C_\bullet(X)\).

8. Worked Examples

Let \(T^2=S^1\times S^1\). Since \(H_i(S^1;\mathbb Z)\cong \mathbb Z\) for \(i=0,1\) and vanishes otherwise, every \(\Tor\) term vanishes. Hence

\[ H_0(T^2;\mathbb Z)\cong \mathbb Z,\qquad H_1(T^2;\mathbb Z)\cong \mathbb Z^2,\qquad H_2(T^2;\mathbb Z)\cong \mathbb Z. \]

Let \(X=\mathbb{RP}^2\) and take coefficients \(\mathbb Z/2\). Since

\[ H_0(X;\mathbb Z)\cong \mathbb Z,\qquad H_1(X;\mathbb Z)\cong \mathbb Z/2,\qquad H_i(X;\mathbb Z)=0 \text{ for } i\ge 2, \]

in degree \(2\) gives

\[ 0 \to H_2(X;\mathbb Z)\otimes \mathbb Z/2 \to H_2(X;\mathbb Z/2) \to \Tor_1^{\mathbb Z}(\mathbb Z/2,\mathbb Z/2) \to 0. \]

Since \(H_2(X;\mathbb Z)=0\) and \(\Tor_1^{\mathbb Z}(\mathbb Z/2,\mathbb Z/2)\cong \mathbb Z/2\), we obtain

\[ H_2(\mathbb{RP}^2;\mathbb Z/2)\cong \mathbb Z/2. \]

Again let \(X=\mathbb{RP}^2\). The cohomological universal coefficient theorem in degree \(2\) gives

\[ 0 \to \Ext_{\mathbb Z}^1(H_1(X;\mathbb Z),\mathbb Z) \to H^2(X;\mathbb Z) \to \Hom(H_2(X;\mathbb Z),\mathbb Z) \to 0. \]

Since \(H_1(X;\mathbb Z)\cong \mathbb Z/2\), \(H_2(X;\mathbb Z)=0\), and \(\Ext_{\mathbb Z}^1(\mathbb Z/2,\mathbb Z)\cong \mathbb Z/2\), it follows that

\[ H^2(\mathbb{RP}^2;\mathbb Z)\cong \mathbb Z/2. \]

Let \(L=L(p,q)\) be a lens space. Its integral homology is

\[ H_0(L;\mathbb Z)\cong \mathbb Z,\qquad H_1(L;\mathbb Z)\cong \mathbb Z/p,\qquad H_2(L;\mathbb Z)=0,\qquad H_3(L;\mathbb Z)\cong \mathbb Z. \]

Therefore gives

\[ H^2(L;\mathbb Z)\cong \Ext_{\mathbb Z}^1(\mathbb Z/p,\mathbb Z)\cong \mathbb Z/p, \]

while \(H^1(L;\mathbb Z)\cong \Hom(\mathbb Z/p,\mathbb Z)=0\).

References

1. Alex Heller, A Simple Proof of the Künneth Theorem, Proceedings of the American Mathematical Society 11 (1960), no. 5, 676-678. DOI: 10.1090/S0002-9939-1960-0144936-2.
2. Charles A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, 1994, Theorem 2.4.7. DOI: 10.1017/CBO9781139644136.
3. J. Michael Boardman, The Eilenberg-Zilber Theorem, course note, Johns Hopkins University, March 30, 2009. https://math.jhu.edu/~jmb/note/eilzil.pdf.