When is a Homology Class Represented by a Submanifold? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-20T04:27:17Z http://mathoverflow.net/feeds/question/21171 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/21171/when-is-a-homology-class-represented-by-a-submanifold When is a Homology Class Represented by a Submanifold? Steve 2010-04-13T01:15:48Z 2012-02-12T00:52:23Z <blockquote> <p><strong>Possible Duplicate:</strong><br> <a href="http://mathoverflow.net/questions/1489/cohomology-and-fundamental-classes" rel="nofollow">Cohomology and fundamental classes</a> </p> </blockquote> <p>Given an oriented manifold $M$ and an oriented submanifold $\phi:N\to M$ we can obtain a homology class <code>$\phi_*[N]\in H_*(M)$</code> where $[N]$ is the fundamental class of $N$. In general, it is not true that every homology class of $M$ can be represented by a submanifold in this manner, however for some special cases it is.</p> <p>For example, for $M$ an oriented (and closed maybe?) 4-manifold every homology class can be represented by a submanifold. Another example is when $M$ an Euclidean configuration space.</p> <p>My questions are:</p> <p>1) Under what circumstances can every homology class of $M$ be represented by a submanifold and</p> <p>2) What are some examples of manifolds who have homology classes not representable in this manner? </p> http://mathoverflow.net/questions/21171/when-is-a-homology-class-represented-by-a-submanifold/21174#21174 Answer by Andy Putman for When is a Homology Class Represented by a Submanifold? Andy Putman 2010-04-13T01:51:28Z 2010-04-13T02:04:44Z <p>A weaker question replaces "homology class of an embedded submanifold" with $f_{\ast}([M])$ for some compact smooth manifold $M$ and an arbitrary continuous map $f:M \rightarrow X$. Once you give up looking at embedded submanifolds, there is also no reason to restrict yourself to $X$ being a manifold.</p> <p>A lot was proven about this by Thom in his classic paper "Quelques propriétés globales des variétés différentiables", which is more famous for containing his work on cobordism theory. A few of the results contained in that paper are as follows.</p> <p>1) Every mod $2$ homology class can be so represented.</p> <p>2) Integrally, it is true for every class in $H_k$ for $k \leq 6$.</p> <p>3) For every $k \geq 7$, there exist polyhedra $X$ and classes in $H_k(X)$ that cannot be so represented.</p> <p>EDIT : One should also remark that the above is germane to the original question too in many cases. Namely, if $X$ is a smooth $n$-manifold and $M$ is a compact smooth $k$-manifold and $f:M \rightarrow X$ is arbitrary, then $f$ is homotopic to an embedding as long as $k &lt; n/2$.</p> http://mathoverflow.net/questions/21171/when-is-a-homology-class-represented-by-a-submanifold/21185#21185 Answer by Don Stanley for When is a Homology Class Represented by a Submanifold? Don Stanley 2010-04-13T06:36:46Z 2010-04-13T06:36:46Z <p>If you really want a submanifold then I guess you can't always do it. For a closed manifold $M$ consider two times the fundamental class $2[M]$. It's easy to see you can't represent this class as a submanifold when $M=S^1$. Perhaps if you take any class in $a\in H_*(M)$ with nonzero self intersection, then $2a$ can't be represented as a submanifold? </p> http://mathoverflow.net/questions/21171/when-is-a-homology-class-represented-by-a-submanifold/21198#21198 Answer by Paul for When is a Homology Class Represented by a Submanifold? Paul 2010-04-13T13:01:38Z 2012-02-12T00:52:23Z <p>Here are a few simple answers to the question you asked:</p> <ol> <li><p>Every class in <code>$H_{n-1}(M;Z)$</code> 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.</p></li> <li><p>Similarly, every class in <code>$H_{n-2}(M;Z)$</code> 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 $CP^{N-1}$.</p></li> <li><p>Transversality says that if you can represent $x\in H_k(M)$ by a <em>map</em> of a smooth manifold (e.g. elements in the image of the Hurewicz map, or by Thom) , and $2k &lt; n$, 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. </p></li> </ol>