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I have posted this problem recently and I think that as it was too long, not many people took interest in it. So I am reposting it also I am giving you an earlier link :I had a problem in manifold which states:

Show that $A^1:Man^{op} \to Mod$ is a functor, where $M$ is a manifold, $A^1(M)$ is the $1$-form on $M$. and for $\phi:M \to N$, $A^1(\phi)=\phi^*$. This means to show that

a) $A^1(M)$ is a module.(What is the ring?)

b) $\phi^*$ preserves module operations(does the ring change?)(I feel yes! but not getting why!!)

c)$(1_M)^*=1_{A^1(M)}$

In part c) I feel that I have to show $w \circ (\Bbb 1_M)_*=(\Bbb 1_{A^1(M)}w$ as $w \circ (\Bbb 1_M)_*=(\Bbb 1_M)^*w$ but how!! Please help me in a), b) and c). Especially I feel that the ring in a) should be $\Bbb R$

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  • $\begingroup$ 1) $\Bbb R$ 2) ?? 3) tautological if you know the definition of pullback $\endgroup$ – user98602 Jan 24 at 15:00
  • $\begingroup$ Sorry, I am messed in the conception. Can you please explain in details? This will be of great help, as manifolds goes above my head. $\endgroup$ – Gimgim Jan 26 at 5:01
  • $\begingroup$ I suggest that you look at a smooth manifolds book, such as John Lee's "Introduction to Smooth Manifolds". $\endgroup$ – user98602 Jan 26 at 17:08
  • $\begingroup$ Yes, I am reading that and Lee and Tu's book but this subject is hard at least to me. So it would of great help if you can expand me. Thanks anyway $\endgroup$ – Gimgim Jan 27 at 2:18

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