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An alternative title is: When can I homotope a continuous map to a smooth immersion?

I have a simple topology problem but it's outside my area of expertise and I worry may be rather subtle. Any help would be appreciated.

The set-up is the following: Let $M$ be some (closed say) $n$ dimensional manifold and suppose that $\Sigma_1$ and $\Sigma_2$ are two closed submanifolds of $M$ of dimension $k$. Note that $\Sigma_1$ and $\Sigma_2$ are allowed to intersect (in my situation they are also embedded but I don't believe this is effects anything). Suppose in addition that $\Sigma_1$ and $\Sigma_2$ are homologous. If $k\leq n-2$ I would like a compact manifold with boundary $\Gamma$ with $\partial \Gamma=\gamma_1\cup \gamma_2$ and a smooth immersion $F:\Gamma\to M$ so that $F(\gamma_i)=\Sigma_i$. In other, words the homology between $\Sigma_1$ and $\Sigma_2$ can be realized by a smooth immersion.

I believe by approximation arguments one can always get a smooth such $F$ without restriction on $k$ but it need not be an immersion (especially if $k=n-1$). My gut is that when you have $k\leq n-2$ since the dimension of the image of $F$ is codimension one you have enough room to perturb it to be an immersion. That is that $F$ is homotopic rel boundary to our desired immersion.

Unfortunately, I don't know enough to formalize this and all my intuition comes from considering curves and domains in $\mathbb{R}^3$ so I'm afraid there may be obstructions in general.

References would be greatly appreciated.

Thanks!

Edit:

As suspected, the question is somewhat subtle . To make it tractable lets assume that $M$ is a $C^\infty$ domain in $\Real^3$ \mathbb{R}^3$(so is a fairly simple three-manifold with boundary) and that the$\Sigma_i$are curves. This is where my intuition says that there should be such a smooth immersion. 2 Clarified Question An alternative title is: When can I homotope a continuous map to a smooth immersion? I have a simple topology problem but it's outside my area of expertise and I worry may be rather subtle. Any help would be appreciated. The set-up is the following: Let$M$be some (closed say)$n$dimensional manifold and suppose that$\Sigma_1$and$\Sigma_2$are two closed submanifolds of$M$of dimension$k$. Note that$\Sigma_1$and$\Sigma_2$are allowed to intersect (in my situation they are also embedded but I don't believe this is effects anything). Suppose in addition that$\Sigma_1$and$\Sigma_2$are homologous. If$k\leq n-2$I would like a compact manifold with boundary$\Gamma$with$\partial \Gamma=\gamma_1\cup \gamma_2$and a smooth immersion$F:\Gamma\to M$so that$F(\gamma_i)=\Sigma_i$. In other, words the homology between$\Sigma_1$and$\Sigma_2$can be realized by a smooth immersion. I believe by approximation arguments one can always get a smooth such$F$without restriction on$k$but it need not be an immersion (especially if$k=n-1$). My gut is that when you have$k\leq n-2$since the dimension of the image of$F$is codimension one you have enough room to perturb it to be an immersion. That is that$F$is homotopic rel boundary to our desired immersion. Unfortunately, I don't know enough to formalize this and all my intuition comes from considering curves and domains in$\mathbb{R}^3$so I'm afraid there may be obstructions in general. References would be greatly appreciated. Thanks! Edit: As suspected, the question is somewhat subtle . To make it tractable lets assume that$M$is a$C^\infty$domain in$\Real^3$(so is a fairly simple three-manifold with boundary) and that the$\Sigma_i$are curves. This is where my intuition says that there should be such a smooth immersion. 1 # Realizing a homology by a smooth immersion An alternative title is: When can I homotope a continuous map to a smooth immersion? I have a simple topology problem but it's outside my area of expertise and I worry may be rather subtle. Any help would be appreciated. The set-up is the following: Let$M$be some (closed say)$n$dimensional manifold and suppose that$\Sigma_1$and$\Sigma_2$are two closed submanifolds of$M$of dimension$k$. Note that$\Sigma_1$and$\Sigma_2$are allowed to intersect (in my situation they are also embedded but I don't believe this is effects anything). Suppose in addition that$\Sigma_1$and$\Sigma_2$are homologous. If$k\leq n-2$I would like a compact manifold with boundary$\Gamma$with$\partial \Gamma=\gamma_1\cup \gamma_2$and a smooth immersion$F:\Gamma\to M$so that$F(\gamma_i)=\Sigma_i$. In other, words the homology between$\Sigma_1$and$\Sigma_2$can be realized by a smooth immersion. I believe by approximation arguments one can always get a smooth such$F$without restriction on$k$but it need not be an immersion (especially if$k=n-1$). My gut is that when you have$k\leq n-2$since the dimension of the image of$F$is codimension one you have enough room to perturb it to be an immersion. That is that$F$is homotopic rel boundary to our desired immersion. Unfortunately, I don't know enough to formalize this and all my intuition comes from considering curves and domains in$\mathbb{R}^3\$ so I'm afraid there may be obstructions in general.

References would be greatly appreciated.

Thanks!