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In exercise 47 from Gauge Fields, Knots and Gravity by Baez and Munain, we want to show that if $\phi:M\to N$ is a map of smooth manifolds, then there is a unique pullback map on forms $$\phi^*:\Omega(N)\to \Omega(M)$$

that agrees with the pullback of 0-forms and 1-forms, is linear, and distributes over the wedge product (I've included an image since the TeX won't render here for some reason): enter image description here

My candidate definition for the pullback of a $k$-form was to say that enter image description here

This seems to make sense, and resembles both the definition above for 1-forms and others I've found in other sources/questions here on MSE. However, I'm confused as to what the second linearity property would mean-- for example, if $\omega$ is a $k$-form and $\mu$ is a $p$-form and $k\neq p$, how can I use this definition when $\omega$ and $\mu$ act on different objects?

Would it be correct to say that elements of $\Omega(N)=\bigoplus_p \Omega^p (N)$ act on the tensor algebra $T(\text{Vect}(N))$ "homogeneously," i.e. so that for example $\omega+\mu$ where $\omega$ is a $k$-form and $\mu$ a $p$-form would act on $\text{Vect}(N)^{\otimes k} \bigoplus \text{Vect}(N)^{\otimes p}$ so that $\omega$ acts on the first summand and $\mu$ on the second?

LeonardoOiler
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    I have never seen the sum of a $k$-form and a $p$-form when $p\not=k,.$ – Kurt G. Sep 28 '23 at 08:46
  • I guess I’m confused about the meaning of the direct sum $\Omega(N)=\bigoplus_p \Omega^p(N).$ $\Omega(N)$ was defined as the algebra generated by locally finite linear combinations of wedge products of 1-forms, so I gathered this meant we could add any two such forms, even if they have different degrees. I guess if one views this direct sum as consisting of tuples of $k$-forms (with $k$ varying) instead, any linear combination of forms of differing degrees (say, an $n$-form and an $m$-form) acts on a pair consisting of an $n$-tuple of vector fields and an $m$-tuple of vector fields? – LeonardoOiler Sep 28 '23 at 22:54
  • (In which case, just focusing on defining pullbacks, exterior derivatives, etc for $k$-forms would suffice for the general case) – LeonardoOiler Sep 28 '23 at 22:58
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    Apparently this direct sum of vector space notation is used. Be it as it may. Now let's put back on our undergraduate hat of linear algebra: any two vector spaces $U,V$ have a direct sum $U\oplus V,.$ This does not mean we are allowed to add $u\in U$ and $v\in V,.$ The issue of adding differential forms with different degree is also discusse in that link. – Kurt G. Sep 28 '23 at 23:17
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    ... any two $K$-vector spaces $U,V$ to be precise. (I don't want any Bourbakist to tell me I have no clue -:) .) – Kurt G. Sep 28 '23 at 23:27
  • Yes, thank you for pointing that out; I think I get what's going on now. I've been writing pretty sloppily-- when I said "linear combination of forms of differing degrees," I really meant e.g. a tuple consisting of an $n$-form and an $m$-form, i.e. an element of $\bigoplus_p \Omega^p(N)$ where only the $n$th and $m$th entries are nonzero. In the definition of Baez and Munain (as an algebra generated by wedges of 1-forms) $\Omega(N)$ really does decompose as a direct sum of subspaces $\Omega^p(N)$, but I think I get now why it's really the $\Omega^p(N)$ that are important, even if one – LeonardoOiler Sep 28 '23 at 23:54
  • could technically consider arbitrary linear combinations. – LeonardoOiler Sep 28 '23 at 23:54
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    I do not know what you want to do with those pullbacks in the future but I highly recommend a few concrete calculations (of surface integrals for example) to see them in action. That answer of Andrew D. Hwang is a real gem. – Kurt G. Sep 29 '23 at 00:39
  • That is a great answer, thanks for the link! I'll definitely do some concrete examples. – LeonardoOiler Sep 29 '23 at 02:06

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