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I am studying the article Immersion Theory for Homotopy theorists by Michael Weiss for my bachelor thesis. The main theorem states that the space of immersions and formal immersions between two smooth manifolds $M$ and $N$ are weakly homotopy equivalent, with the weak equivalence given by the map $$ \text{imm}(M,N)\to \text{fimm}(M,N),\quad f\mapsto (f,\mathrm{d}f). $$

In the article he never mentions continuity of this map, and maybe it's really trivial, but I'm having some difficulty proving this. For the first component this is almost by definition of the $C^\infty$ topology, but for the second component I can't work it out.

Could anyone help me out? I also posted this to math.stackexchange as I don't really have a clue what level this type of material is. Feel free to delete the question If it doesn't belong here.

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  • $\begingroup$ If you use the $C^k$-topology on both spaces, it is continuous for all $k \geq 1$. Which definition of the $C^k$-topology are you using? There are a variety of equivalent definitions. Hirsch's book is quite good about running through the equivalent definitions. For example, the simplest argument involves the $C^1$-topology which you could describe as the uniform topology when you restrict the derivative of an immersion to its unit tangent bundle. In that regard, your topology on your domain and codomain are given by essentially the same metric. This of course assumes $M$ compact. $\endgroup$
    – Ryan Budney
    Commented Jul 14, 2023 at 5:52
  • $\begingroup$ Perhaps that does not fully answer your question, but it should get you started. Do take a look at Hirsch's book on Differential Topology. $\endgroup$
    – Ryan Budney
    Commented Jul 14, 2023 at 5:53
  • $\begingroup$ @RyanBudney I have already taken a look at Hirsch's book, and what I get from the article, Weiss uses compact-open (what Hirsch calls weak) $C^\infty$ topology on the space of immersions, whereas the topology on the space of formal immersions results from the subspace of $C^0(M,N)\times C^0(TM,f^*TN)$ where both spaces in the product are equipped with the compact open (weak) C^0 topology. I don't believe we are allowed to assume $M$ compact in this case, so that means it matters if we work with the strong or weak topology, right? $\endgroup$
    – heervande
    Commented Jul 14, 2023 at 10:13
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    $\begingroup$ Sure, but this isn't a forum for grad student homework problems. I'm just trying to give you a sense for a first approach. Once you see a way to prove a theorem in a special case it's often fairly easy to extrapolate to full-generality. $\endgroup$
    – Ryan Budney
    Commented Jul 14, 2023 at 16:49
  • $\begingroup$ Yes, I understand that. The reason I specified the topology on $\text{fimm}(M,N)$ is because of your remark that this map is continuous for $k\geq 1$. Is it not continuous for $k=0$ then? In other words, I'm not asking for a solution but rather whether if I have the spaces defined correctly like this. $\endgroup$
    – heervande
    Commented Jul 14, 2023 at 19:52

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