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In Lee's Introduction to Manifolds book in chapter 13 (page 335) they define a contraction $i_X: \Lambda^k(V) \to \Lambda^{k-1}(V)$ as $$i_X \omega (Y_1,...,Y_{k-1}) = \omega(X, Y_1,...,Y_{k-1})$$

First question I have is: Isn't there a typo and shouldn't it be $i_X: \Lambda^{k-1}(V) \to \Lambda^k(V)$?

My second, and more important question is: if $\omega$ is a $k$ form then the LHS $\omega (Y_1,...,Y_{k-1})$ doesn't make sense since it's missing a vector field. On the other hand, if $\omega$ is a $k-1$ form then on the RHS $\omega(X, Y_1,...,Y_{k-1})$ doesn't make sense cause there is an extra vector field.

Math_Day
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    $i_X\omega$ is a map whose arguments on the left-hand side are $Y_1, \ldots, Y_{k - 1}$, so it's an element of $\bigwedge^{k - 1}(V)$. – Travis Willse Jul 16 '23 at 18:18

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You’re reading it wrong, there’s no mistake. Given a $k$-form $\omega$, and a vector field $X$, one defines a $(k-1)$-form, $\iota_X\omega$, called the interior product of $\omega$ with $X$. The value of $\iota_X\omega$ on $k-1$ vector fields $Y_1,\dots, Y_{k-1}$ is defined to be the value of $\omega$ on the $k$ vector fields $X,Y_1,\dots, Y_{k-1}$. In symbols, \begin{align} (\iota_X\omega)(Y_1,\dots, Y_{k-1})&:=\omega(X,Y_1,\dots, Y_{k-1}). \end{align}

You seem to be reading things as $\iota_X\bigg(\omega(Y_1,\dots, Y_{k-1})\bigg)$, which is not what was intended, and is clearly nonsensical for the reason you mentioned.

peek-a-boo
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