4 fixed spelling of Hurewicz

1. Every class in $H_{n-1}(M;Z)$ for $M$ orientable is represented by a submanifold: choose a smooth map $f:M\to S^1$ representing the Poincare dual in $H^1(M;Z)=[M,S^1]$ and take the preimage of a point. In dimensions>2 it can be taken connected.

2. Similarly, every class in $H_{n-2}(M;Z)$ for $M$ orientable is represented by a submanifold: choose a smooth map $f:M\to CP^\infty$ representing the Poincare dual in $H^2(M;Z)=[M,CP^\infty]$, homotop $f$ into a finite skeleton, say $CP^N$, and take the preimage of $CP^{N-1}$.

3. Transversality says that if you can represent $x\in H_k(M)$ by a map of a smooth manifold (e.g. elements in the image of the Hurewitz Hurewicz map, or by Thom) , and $2k < n$, then you can represent it by an embedded submanifold (as Andy mentions above). For example, any class in $H_1(M)$ for $dim(M)\ge 3$. With care you can also make this work for $2k=n$, and there are techniques available in the "metastable" range (no triple points) involving generalizations of Whitney's trick and other ways to replace double points.

3 fixed typos

1. Every class in $H_{n-1}(M;Z)$ for $M$ orientable is represented by a submanifold: choose a smooth map $f:M\to S^1$ representing the Poincare dual in $H^1(M;Z)=[M,S^1]$ and take the preimage of a point. In dimensions>2 it can be taken connected.
2. Similarly, every class in $H_{n-2}(M;Z)$ for $M$ orientable is represented by a submanifold: choose a smooth map $f:M\to CP^\infty$ representing the Poincare dual in $H^1(M;Z)=[M,CP^\infty]$, H^2(M;Z)=[M,CP^\infty]$, homotop$f$into a finite skeleton, say$CP^N$, and take the preimage of$CP^{N-1}$. 3. Transversality says that if you can represent$x\in H_k(M)$by a map of a smooth manifold (e.g. elements in the image of the Hurewitz map, or by Thom) , and$2k < n$, then you can represent it by an embedded submanifold (as Andy mentions above). For example, any class in$H_1(M)$for$dim(M)\ge 3$. With care you can also make this work for$2k=n$, and there are techniques available in the "metastable" range (no triple points) involving generalizations of Whitney's trick and other ways to replace double points. 2 added 191 characters in body Here are a few simple answers to the question you asked: 1. Every class in$H_{n-1}(M;Z)$for$M$orientable is represented by a submanifold: choose a smooth map$f:M\to S^1$representing the Poincare dual in$H^1(M;Z)=[M,S^1]$and take the preimage of a point. In dimensions>2 it can be taken connected. 2. Similarly, every class in$H_{n-2}(M;Z)$for$M$orientable is represented by a submanifold: choose a smooth map$f:M\to CP^\infty$representing the Poincare dual in$H^1(M;Z)=[M,CP^\infty]$, homotop$f$into a finite skeleton, say$CP^N$, and take the preimage of$CP^{N-1}$. 3. Transversality says that if you can represent$x\in H_k(M)$by a map of a smooth manifold (e.g. elements in the image of the Hurewitz map, or by Thom) , and$2k < n$, then you can represent it by an embedded submanifold (as Andy mentions above). For example, any class in$H_1(M)$for$dim(M)\ge 3$. With care you can also make this work for$2k=n$. 2k=n$, and there are techniques available in the "metastable" range (no triple points) involving generalizations of Whitney's trick and other ways to replace double points.