This is probably trivial but I am trying to find whether or not the degree of a map $f:X\to Y$, with $X$ and $Y$ compact oriented manifolds can be extended to the case where $Y$ is non compact like $GL(N,\mathbb{C})$. In particular I am dealing with $f:S^1\to GL(N,\mathbb{C})$ and it seems to me that since $\pi_1(GL(N,\mathbb{C}))\cong \pi_1(U(N)) = \mathbb{Z}$, the degree should exist and be an integer. Am I wrong? However, I also know that it must be deg$(f) \int_Y\omega=\int_X f^*\omega$, with $\omega$ an invariant differential form and $f^*\omega$ the pull-back of $\omega$. I could choose $\omega = g^{-1}dg$ the Maurer-Cartan form since I am working with a Lie group but the fact that it is non compact means that its volume is infinite. What is the correct way of computing the degree in this case?
I apologize in advance if I am saying something wrong but I am new to some of these concepts. Any kind of help is appreciated!
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I actually found a reference that talks about maps $f:S^{n-1}\to GL(N,\mathbb{C})$ in an old paper by Atiyah https://aareyanmanzoor.github.io/assets/images/atiyah1967.pdf where he says that for $n$ even we can define an integer called deg$f$ (Bott's theorem). This is explained at the bottom of the second page (238) and in the following page he mentions
There is also a differential definition of deg$f$. We put deg$f = \int_{S^{n-1}}f^*\omega$, where $\omega$ is a certain explicitly defined invariant differential form on $GL(N,\mathbb{C})$ and $f^*\omega$ is the induced form on $S^{n-1}$.
Doesn't this mean that the invariant I am looking for is indeed the degree of that map? Can $\omega$ be the Maurer-Cartan form?
