Differentiable structures

I am taking a course on differential topology this semester and will be writing a series of posts on this topic. In this post, we will look at the notion of differentiable structures on a topological manifold.

Differentiable structure on a manifold is a basic setting in which we can talk of differentiability of functions defined on a manifold. To do this we assume the standard differentiable structure of \mathbb{R}^n i.e., we assume the notions of differentiability of functions from \mathbb{R}^n \to \mathbb{R}^m. To start with, n-manifolds are spaces which locally look like real vector space \mathbb{R}^n. More formally, a n-dimensional manifold M^n is defined by the following structure:

1) A topological space M=\bigcup_{\alpha}U_{\alpha} where U_{\alpha} is collection of open sets covering M.( Sometimes extra conditions of hausdorffness and second countability are imposed on the space. We don’t do it here.)

2) An atlas of charts \phi_{\alpha}:U_{\alpha}\to V_{\alpha} which are homeomorphisms from U_{\alpha} in M to V_{\alpha} in \mathbb{R}^n s.

Using these charts we can identify the points in the space with points in \mathbb{R}^n locally. However if we two different charts overlap same point in the space may be mapped to different points in \mathbb{R}^n in different charts.

Now we go to the differentiable structure on manifolds given which we can speak of  differentiability, derivatives etc. The differentiable structure is defined as follows

A function f:U_{\alpha}\to \mathbb{R} is said to be differentiable if f\circ\phi_\alpha^-1 is differentiable. That we require the function obtained by pulling back using \phi_{\alpha} to be differentiable.(These are just functions from \mathbb{R}to\mathbb{R}.We know what differentiability of these functions mean.) So we now have a class of differentiable functions on U_{\alpha},  D_\alpha = \{f:U_{\alpha}\to \mathbb{R} \mbox{ differentiable wrt to }\phi_\alpha \}.

Hence we have defined the local notion of differentiability (We can speak of functions being differentiable with respect to a particular chart). To speak of differentiability globally ( functions being differentiable on the manifold) we need these local notions to be consistent. For any \alpha,\beta such that U_{\alpha}\cap U_{\beta}\ne \phi we need that D_{\alpha}| U_{\alpha}\cap U_{\beta}=D_{\beta}| U_{\alpha}\cap U_{\beta}. That is, we require every function on the intersection obtained by restriction, to be differentiable in both \alpha notion and \beta notion of differentiability or not differentiable in both. Said otherwise, we don’t want to have a function on the same region to be differentiable in\alpha chart and not differentiable in the \beta chart or vice versa. So thus imposing the consistency conditions we have a differentiable structure on our manifold.

Note that the compatibility of charts above is defined in terms of set of differentiable functions on the chart. But one can equivalently state the compatibility merely in terms of the charts \phi_\alpha, \phi_\beta. We require the transition functions \phi_\alpha\circ\phi_\beta^-1 and \phi_\beta\circ\phi_\alpha^-1 to be differentiable. It can be easily checked.

To summarise, differential structure is given by local notions which are turned into a global notion by imposing a compatibility through differentiable transition functions. One can similarly define the C^k-differentiable structures, analytic structure  on the manifold by requiring an atlas of compatible charts with transition functions being k times continuously differentiable, analytic respectively. One could also define similar structures with complex manifolds where the space looks like complex vector space locally. We can talk of complex analytic structures.

We have a defined what we mean by a differentiable structure. Now we can ask if every topological manifold admits a differentiable structure. If it does, is it unique?  If there are more, how different are these structures?  We can decide to consider some types of structures to be equivalent and can ask how many different structures exist when we quotient out the equivalences.

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