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This is an answer to 1. It is an edit of a previous answer based on the incorrect assumption that $\mathbb Z/k$-manifold means a closed manifold with a $\mathbb Z/k$-orientation. Thanks to Danny Ruberman for pointing out in the comments that a $\mathbb Z/k$-manifold is a type of manifold with singularities introduced by Morgan and Sullivan in The transversality characteristic class and linking cycles in surgery theory.

Roughly, a $\mathbb Z/k$-manifold of dimension $n$ is obtained from an oriented $n$-manifold with boundary whose boundary is partitioned into $k$ disjoint isomorphic closed $(n-1)$-manifolds by identifying the boundary pieces by orientation preserving diffeomorphisms. See section 2 of $\mathbb Z_k$-stratifolds by Andrés Angel, Carlos Segovia and Arley Fernando Torres for a more precise definition.

A $\mathbb Z/k$-manifold has a fundamental class in mod $k$ homology whose Bockstein is an integral fundamental class for the codimension 1 submanifold along which it is singular.

Given a CW complex $X$, any element of $H_n(X;\mathbb Z/2^k)$ is the image of the fundamental class of a $\mathbb Z/2^k$-manifold $M$ under some map $f \colon M \to X$. Here is a (possibly overcomplicated) justification:

Morgan and Sullivan show (or at least state, bottom of p. 471) that the bordism of $\mathbb Z/k$-manifolds is the homology theory represented by $MSO\wedge M\mathbb Z/k$ classifying cobordism with $\mathbb Z/k$-coefficients.

The spectrum $MSO_{(2)}$ is a wedge of suspensions of $H\mathbb Z/2$ and $H\mathbb Z_{(2)}$ and the canonical map $MSO_{(2)} \to H\mathbb Z_{(2)}$ splits. Smashing with the Moore spectrum $M\mathbb Z/2^k$ we see that the same is true of the map $MSO\wedge M\mathbb Z/2^k \to H\mathbb Z/2^k$. The (split surjective) induced map on homology sends a bordism class $[f \colon M \to X]$ to the image $f_*[M] \in H_*(X;\mathbb Z/2^k)$ of the fundamental class of the $\mathbb Z/2^k$-manifolds $M$. This completes the proof.

Note that this argument would not work for mod $l$ homology with $l$ not a power of $2$. See the paper by Angel, Segovia and Torres linked to above for more information and examples of $p$-torsion classes (with $p$ an odd prime) not represented by maps from $\mathbb Z/p$-manifolds.

Finally, see also Proposition 5.7.4 in Gompf and Stipsicz, 4-manifolds and Kirby calculus, for the original Hirzebruch–Hopf proof as well as a description of the Teichner–Vogt proof in All 4-manifolds have $\operatorname{Spin}^c$-structures for arbitrary oriented $4$-manifolds linked to in the comments on the question (1 2 3). Edit: See also an interesting recent proof of LeBrun's in Twistors, Self-Duality and Spin^c structures which is quite different from the proofs above. As written it only applies to compact $4$-manifolds but it is not hard to slightly adapt it so that it works for all $4$-manifolds.

This is an answer to 1. It is an edit of a previous answer based on the incorrect assumption that $\mathbb Z/k$-manifold means a closed manifold with a $\mathbb Z/k$-orientation. Thanks to Danny Ruberman for pointing out in the comments that a $\mathbb Z/k$-manifold is a type of manifold with singularities introduced by Morgan and Sullivan in The transversality characteristic class and linking cycles in surgery theory.

Roughly, a $\mathbb Z/k$-manifold of dimension $n$ is obtained from an oriented $n$-manifold with boundary whose boundary is partitioned into $k$ disjoint isomorphic closed $(n-1)$-manifolds by identifying the boundary pieces by orientation preserving diffeomorphisms. See section 2 of $\mathbb Z_k$-stratifolds by Andrés Angel, Carlos Segovia and Arley Fernando Torres for a more precise definition.

A $\mathbb Z/k$-manifold has a fundamental class in mod $k$ homology whose Bockstein is an integral fundamental class for the codimension 1 submanifold along which it is singular.

Given a CW complex $X$, any element of $H_n(X;\mathbb Z/2^k)$ is the image of the fundamental class of a $\mathbb Z/2^k$-manifold $M$ under some map $f \colon M \to X$. Here is a (possibly overcomplicated) justification:

Morgan and Sullivan show (or at least state, bottom of p. 471) that the bordism of $\mathbb Z/k$-manifolds is the homology theory represented by $MSO\wedge M\mathbb Z/k$ classifying cobordism with $\mathbb Z/k$-coefficients.

The spectrum $MSO_{(2)}$ is a wedge of suspensions of $H\mathbb Z/2$ and $H\mathbb Z_{(2)}$ and the canonical map $MSO_{(2)} \to H\mathbb Z_{(2)}$ splits. Smashing with the Moore spectrum $M\mathbb Z/2^k$ we see that the same is true of the map $MSO\wedge M\mathbb Z/2^k \to H\mathbb Z/2^k$. The (split surjective) induced map on homology sends a bordism class $[f \colon M \to X]$ to the image $f_*[M] \in H_*(X;\mathbb Z/2^k)$ of the fundamental class of the $\mathbb Z/2^k$-manifolds $M$. This completes the proof.

Note that this argument would not work for mod $l$ homology with $l$ not a power of $2$. See the paper by Angel, Segovia and Torres linked to above for more information and examples of $p$-torsion classes (with $p$ an odd prime) not represented by maps from $\mathbb Z/p$-manifolds.

Finally, see also Proposition 5.7.4 in Gompf and Stipsicz, 4-manifolds and Kirby calculus, for the original Hirzebruch–Hopf proof as well as a description of the Teichner–Vogt proof in All 4-manifolds have $\operatorname{Spin}^c$-structures for arbitrary oriented $4$-manifolds linked to in the comments on the question (1 2 3). Edit: See also an interesting recent proof of LeBrun's in Twistors, Self-Duality and Spin^c structures which is quite different from the proofs above. As written it only applies to compact $4$-manifolds but it is not hard to slightly adapt it so that it works for all $4$-manifolds.

This is an answer to 1. It is an edit of a previous answer based on the incorrect assumption that $\mathbb Z/k$-manifold means a closed manifold with a $\mathbb Z/k$-orientation. Thanks to Danny Ruberman for pointing out in the comments that a $\mathbb Z/k$-manifold is a type of manifold with singularities introduced by Morgan and Sullivan in The transversality characteristic class and linking cycles in surgery theory.

Roughly, a $\mathbb Z/k$-manifold of dimension $n$ is obtained from an oriented $n$-manifold with boundary whose boundary is partitioned into $k$ disjoint isomorphic closed $(n-1)$-manifolds by identifying the boundary pieces by orientation preserving diffeomorphisms. See section 2 of $\mathbb Z_k$-stratifolds by Andrés Angel, Carlos Segovia and Arley Fernando Torres for a more precise definition.

A $\mathbb Z/k$-manifold has a fundamental class in mod $k$ homology whose Bockstein is an integral fundamental class for the codimension 1 submanifold along which it is singular.

Given a CW complex $X$, any element of $H_n(X;\mathbb Z/2^k)$ is the image of the fundamental class of a $\mathbb Z/2^k$-manifold $M$ under some map $f \colon M \to X$. Here is a (possibly overcomplicated) justification:

Morgan and Sullivan show (or at least state, bottom of p. 471) that the bordism of $\mathbb Z/k$-manifolds is the homology theory represented by $MSO\wedge M\mathbb Z/k$ classifying cobordism with $\mathbb Z/k$-coefficients.

The spectrum $MSO_{(2)}$ is a wedge of suspensions of $H\mathbb Z/2$ and $H\mathbb Z_{(2)}$ and the canonical map $MSO_{(2)} \to H\mathbb Z_{(2)}$ splits. Smashing with the Moore spectrum $M\mathbb Z/2^k$ we see that the same is true of the map $MSO\wedge M\mathbb Z/2^k \to H\mathbb Z/2^k$. The (split surjective) induced map on homology sends a bordism class $[f \colon M \to X]$ to the image $f_*[M] \in H_*(X;\mathbb Z/2^k)$ of the fundamental class of the $\mathbb Z/2^k$-manifolds $M$. This completes the proof.

Note that this argument would not work for mod $l$ homology with $l$ not a power of $2$. See the paper by Angel, Segovia and Torres linked to above for more information and examples of $p$-torsion classes (with $p$ an odd prime) not represented by maps from $\mathbb Z/p$-manifolds.

Finally, see also Proposition 5.7.4 in Gompf and Stipsicz, 4-manifolds and Kirby calculus, for the original Hirzebruch–Hopf proof as well as a description of the Teichner–Vogt proof in All 4-manifolds have $\operatorname{Spin}^c$-structures for arbitrary oriented $4$-manifolds linked to in the comments on the question (1 2 3).

This is an answer to 1. It is an edit of a previous answer based on the incorrect assumption that $\mathbb Z/k$-manifold means a closed manifold with a $\mathbb Z/k$-orientation. Thanks to Danny Ruberman for pointing out in the comments that a $\mathbb Z/k$-manifold is a type of manifold with singularities introduced by Morgan and Sullivan in The transversality characteristic class and linking cycles in surgery theory.

Roughly, a $\mathbb Z/k$-manifold of dimension $n$ is obtained from an oriented $n$-manifold with boundary whose boundary is partitioned into $k$ disjoint isomorphic closed $(n-1)$-manifolds by identifying the boundary pieces by orientation preserving diffeomorphisms. See section 2 of $\mathbb Z_k$-stratifolds by Andrés Angel, Carlos Segovia and Arley Fernando Torres for a more precise definition.

A $\mathbb Z/k$-manifold has a fundamental class in mod $k$ homology whose Bockstein is an integral fundamental class for the codimension 1 submanifold along which it is singular.

Given a CW complex $X$, any element of $H_n(X;\mathbb Z/2^k)$ is the image of the fundamental class of a $\mathbb Z/2^k$-manifold $M$ under some map $f \colon M \to X$. Here is a (possibly overcomplicated) justification:

Morgan and Sullivan show (or at least state, bottom of p. 471) that the bordism of $\mathbb Z/k$-manifolds is the homology theory represented by $MSO\wedge M\mathbb Z/k$ classifying cobordism with $\mathbb Z/k$-coefficients.

The spectrum $MSO_{(2)}$ is a wedge of suspensions of $H\mathbb Z/2$ and $H\mathbb Z_{(2)}$ and the canonical map $MSO_{(2)} \to H\mathbb Z_{(2)}$ splits. Smashing with the Moore spectrum $M\mathbb Z/2^k$ we see that the same is true of the map $MSO\wedge M\mathbb Z/2^k \to H\mathbb Z/2^k$. The (split surjective) induced map on homology sends a bordism class $[f \colon M \to X]$ to the image $f_*[M] \in H_*(X;\mathbb Z/2^k)$ of the fundamental class of the $\mathbb Z/2^k$-manifolds $M$. This completes the proof.

Note that this argument would not work for mod $l$ homology with $l$ not a power of $2$. See the paper by Angel, Segovia and Torres linked to above for more information and examples of $p$-torsion classes (with $p$ an odd prime) not represented by maps from $\mathbb Z/p$-manifolds.

Finally, see also Proposition 5.7.4 in Gompf and Stipsicz, 4-manifolds and Kirby calculus, for the original Hirzebruch–Hopf proof as well as a description of the Teichner–Vogt proof in All 4-manifolds have $\operatorname{Spin}^c$-structures for arbitrary oriented $4$-manifolds linked to in the comments on the question (1 2 3). Edit: See also an interesting recent proof of LeBrun's in Twistors, Self-Duality and Spin^c structures which is quite different from the proofs above. As written it only applies to compact $4$-manifolds but it is not hard to slightly adapt it so that it works for all $4$-manifolds.

This is an answer to 1. It is an edit of a previous answer based on the incorrect assumption that $\mathbb Z/k$-manifold means a closed manifold with a $\mathbb Z/k$-orientation. Thanks to Danny Ruberman for pointing out in the comments that a $\mathbb Z/k$-manifold is a type of manifold with singularities introduced by Morgan and Sullivan in The transversality characteristic class and linking cycles in surgery theory.

Roughly, a $\mathbb Z/k$-manifold of dimension $n$ is obtained from an oriented $n$-manifold with boundary whose boundary is partitioned into $k$ disjoint isomorphic closed $(n-1)$-manifolds by identifying the boundary pieces by orientation preserving diffeomorphisms. See section 2 of this preprint by Andrés Angel, Carlos Segovia and Arley Fernando Torres for a more precise definition.

A $\mathbb Z/k$-manifold has a fundamental class in mod $k$ homology whose Bockstein is an integral fundamental class for the codimension 1 submanifold along which it is singular.

Given a CW complex $X$, any element of $H_n(X;\mathbb Z/2^k)$ is the image of the fundamental class of a $\mathbb Z/2^k$-manifold $M$ under some map $f \colon M \to X$. Here is a (possibly overcomplicated) justification:

Morgan and Sullivan show (or at least state, bottom of p. 471) that the bordism of $\mathbb Z/k$-manifolds is the homology theory represented by $MSO\wedge M\mathbb Z/k$ classifying cobordism with $\mathbb Z/k$-coefficients.

The spectrum $MSO_{(2)}$ is a wedge of suspensions of $H\mathbb Z/2$ and $H\mathbb Z_{(2)}$ and the canonical map $MSO_{(2)} \to H\mathbb Z_{(2)}$ splits. Smashing with the Moore spectrum $M\mathbb Z/2^k$ we see that the same is true of the map $MSO\wedge M\mathbb Z/2^k \to H\mathbb Z/2^k$. The (split surjective) induced map on homology sends a bordism class $[f \colon M \to X]$ to the image $f_*[M] \in H_*(X;\mathbb Z/2^k)$ of the fundamental class of the $\mathbb Z/2^k$-manifolds $M$. This completes the proof.

Note that this argument would not work for mod $l$ homology with $l$ not a power of $2$. See the paper by Angel, Segovia and Torres linked to above for more information and examples of $p$-torsion classes (with $p$ an odd prime) not represented by maps from $\mathbb Z/p$-manifolds.

Finally, see also Proposition 5.7.4 in Gompf and Stipsicz, 4-manifolds and Kirby calculus, for the original Hirzebruch–Hopf proof as well as a description of the Teichner–Vogt proof in All 4-manifolds have $\operatorname{Spin}^c$-structures for arbitrary oriented $4$-manifolds linked to in the comments on the question (1 2 3).

This is an answer to 1. It is an edit of a previous answer based on the incorrect assumption that $\mathbb Z/k$-manifold means a closed manifold with a $\mathbb Z/k$-orientation. Thanks to Danny Ruberman for pointing out in the comments that a $\mathbb Z/k$-manifold is a type of manifold with singularities introduced by Morgan and Sullivan in The transversality characteristic class and linking cycles in surgery theory.

Roughly, a $\mathbb Z/k$-manifold of dimension $n$ is obtained from an oriented $n$-manifold with boundary whose boundary is partitioned into $k$ disjoint isomorphic closed $(n-1)$-manifolds by identifying the boundary pieces by orientation preserving diffeomorphisms. See section 2 of $\mathbb Z_k$-stratifolds by Andrés Angel, Carlos Segovia and Arley Fernando Torres for a more precise definition.

A $\mathbb Z/k$-manifold has a fundamental class in mod $k$ homology whose Bockstein is an integral fundamental class for the codimension 1 submanifold along which it is singular.

Given a CW complex $X$, any element of $H_n(X;\mathbb Z/2^k)$ is the image of the fundamental class of a $\mathbb Z/2^k$-manifold $M$ under some map $f \colon M \to X$. Here is a (possibly overcomplicated) justification:

Morgan and Sullivan show (or at least state, bottom of p. 471) that the bordism of $\mathbb Z/k$-manifolds is the homology theory represented by $MSO\wedge M\mathbb Z/k$ classifying cobordism with $\mathbb Z/k$-coefficients.

The spectrum $MSO_{(2)}$ is a wedge of suspensions of $H\mathbb Z/2$ and $H\mathbb Z_{(2)}$ and the canonical map $MSO_{(2)} \to H\mathbb Z_{(2)}$ splits. Smashing with the Moore spectrum $M\mathbb Z/2^k$ we see that the same is true of the map $MSO\wedge M\mathbb Z/2^k \to H\mathbb Z/2^k$. The (split surjective) induced map on homology sends a bordism class $[f \colon M \to X]$ to the image $f_*[M] \in H_*(X;\mathbb Z/2^k)$ of the fundamental class of the $\mathbb Z/2^k$-manifolds $M$. This completes the proof.

Note that this argument would not work for mod $l$ homology with $l$ not a power of $2$. See the paper by Angel, Segovia and Torres linked to above for more information and examples of $p$-torsion classes (with $p$ an odd prime) not represented by maps from $\mathbb Z/p$-manifolds.

Finally, see also Proposition 5.7.4 in Gompf and Stipsicz, 4-manifolds and Kirby calculus, for the original Hirzebruch–Hopf proof as well as a description of the Teichner–Vogt proof in All 4-manifolds have $\operatorname{Spin}^c$-structures for arbitrary oriented $4$-manifolds linked to in the comments on the question (1 2 3).