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Let $M$ be a smooth manifold. The classical construction is the tangent bundle $TM$. What does the tangent groupoid $GM$ give me that this construction doesn't, and why is it useful in non-commutative geometry?

Heuristically, the tangent groupoid, which actually is a bundle too, thickens the tangent bundle with approximations to it; instead of being over $M$, it is over $M\mathbb{R}$, and the fibre over zero is the standard tangent bundle.

To construct it, we first note two simpler constructions:

a. Any bundle $E \rightarrow M$ can be considered as a groupoid $G$ with object space $M$ and morphisms $G(a,a)=E_a$ with the obvious composition and all other hom-spaces empty.

b. The pair groupoid $M \times M$ on $M$ has object space $M$ and morphism space $M^2$, with composition $(a,b)(b,c):=(a,c)$ (with all others empty).

Then given a manifold $M$, its tangent groupoid $GM$ is a disjoint union of groupoids $G_t M$ for $t$ in the real line; and where $G_0 M$=$TM$ considered as a groupoid and $G_t M=M \times M$

Now the object space of $GM$ is the disjoint union of the object space of each fibre which is $M$, for each $t$ in the real line. This means we can identify the object space with $M \mathbb{R}$.

We give it a weak topology so that fibres spaces away from the fibre over zero are seen as spaces of approximate tangent vectors, that is:

for $ f \in C^{\infty}M$ we take the weakest topology (the one with the fewest open sets) such that the following are continuous:

a. $(X,m,0) \rightarrow Xf$

b. $(m,n,t) \rightarrow \frac{(fn-fm)}{t}$

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    $\begingroup$ I disagree that the tangent bundle is "the" natural construction. It is so classical! How can a classical construction be "the" natural construction when nature is quantum? $\endgroup$
    – Qiaochu Yuan
    Commented Jul 9, 2012 at 0:13
  • $\begingroup$ You have a point. I meant natural as in what I would naturally reach for when I want to think about tangents. Classical is a better & right word. $\endgroup$
    – Mozibur Ullah
    Commented Jul 9, 2012 at 0:45
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    $\begingroup$ @Qiaochu: In mathematics when people talk about a natural construction, they mean it is immediately defined from the very core nature of the previous construction (and in most cases) it does not depend on specific choice of other object, for example, the quotient map in algebraic construction. So, we should not expect a notion be inspired by nature in order to call it natural. $\endgroup$
    – user23860
    Commented Jul 9, 2012 at 3:04
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    $\begingroup$ @Mozibur Ullah: Maybe you could edit your question and add a brief explanation what this tangent groupoid is. Just in case that somebody doesn't have Connes' books at hand... $\endgroup$
    – Konrad Waldorf
    Commented Jul 9, 2012 at 13:27
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    $\begingroup$ @Vahid: it was something of a joke. Rest assured I am well aware of what "natural" means, but I had some vague idea that the tangent groupoid was relevant to noncommutative geometry and so I thought that the use of the word "natural" in this context to describe a commutative construction was funny. $\endgroup$
    – Qiaochu Yuan
    Commented Jul 9, 2012 at 14:59

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The tangent groupoid can be used in constructing the index map and proving the Atiyah-Singer index theorem. This may illustrate its importance. Higson and Roe will write a book on it.

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    $\begingroup$ I wouldn't gamble on that book appearing anytime soon - their progress seems to have stalled about four or five years ago. Nevertheless you can find preliminary yet well developed versions of it online. There is also Higson's paper "The Tangent Groupoid and the Index Theorem" which uses a similar approach but without asymptotic morphisms. $\endgroup$
    – Paul Siegel
    Commented Jun 22, 2013 at 18:48

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