# What does having a bar on a manifold mean?

If $$M$$ is a manifold then what is denoted as $$\overline{M}$$?

I am guessing that it means a reversal of orientation.

Related to the above I would like to understand the following construction of a $$d$$-dimensional Riemannian category that I recently came across:

Objects are $$(d−1)$$ Riemannian manifolds. Morphisms between two oriented $$(d − 1)$$-dimensional Riemannian manifolds $$N_1$$ and $$N_2$$ are oriented $$d$$-dimensional Riemannian manifolds $$M$$, such that $$\partial M = N_1 ⊔ \overline{N_2}$$. The orientation on all three manifolds should naturally agree, and the metric on $$M$$ agrees with the metric on $$N_1$$ and $$N_2$$ on a collar of the boundary. The composition is the gluing of such Riemannian cobordisms.

I would like to know more about this intuition of defining a morphism between two manifolds as another manifold itself! One is used to thinking of morphisms as maps.

• Yes, your guess is correct. Maybe you should get used to the fact that it's not always a good idea to think of morphisms as functions or maps. But maybe it helps to think of this rather as an equivalence relation.
– t.b.
Commented May 15, 2011 at 20:02
• The bar notation is used for orientation reversal, especially in the context where the authors like to mention complex projective spaces. I never see $-\mathbb CP^n$ in the literature, it's always $\overline{\mathbb CP^n}$. The category you're talking about is a standard cobordism category construction. In a sense it goes back to Thom and Pontriagin but they didn't want to think of it as a category. Commented May 15, 2011 at 20:09
• This is about latex in your post. I made some edits, but they won't show up until they're "peer-reviewed". \overline{} tends to look better than \bar{}. Compare $\overline{M}$ and $\bar{M}$.
– Josh
Commented May 15, 2011 at 20:12
• @gary: You can delete your own comments by clicking on the little cross that appears when you hover over the comment. Commented May 16, 2011 at 1:39

I would like to know more about this intuition of defining a morphism between two manifolds as another manifold itself! One is used to thinking of morphisms as maps.

One intuition is from physics: one thinks of $n$-manifolds as "space," and of $n+1$-dimensional cobordisms between them as "time evolution." A nice introduction to these ideas is Baez's Physics, Topology, Logic and Computation: a Rosetta Stone.

• @Qiacu Thanks for the reference. The article seems accessible to first-timers.right? Or is there some specialized pre-requisite? Commented May 17, 2011 at 7:33
• I think it's fairly accessible. Commented May 17, 2011 at 7:35

You've simply encountered the definition of a topological quantum field theory (in the formulation gave by M. Atiyah). I can provide you lots of documentation about that, but I'm a little hurried now... I'll edit this post during the day. For the moment I'm copypasting this:

Topological Quantum Field Theory, or TQFT for short, is a notion which originally arose from ideas by E. Witten in quantum physics (SUSY-Quantum Mechanics). Since then it has developed considerably in a number of directions, and in particular has had a pervasive influence in algebraic topology, by means of its capacity of computing algebraic invariants of a suitable class of manifolds, using so-called "cobordisms" between closed manifolds (smooth manifolds having them as boundaries).

Atiyah gave a first axiomatization of Witten's theory in his paper [1]; I'll try to focus on the illuminating analogy (duality) beween Atiyah's axioms for TQFT and the Eilenberg-Steenrod axioms for homology as in [2]. Atiyah's approach is implicitly categorical; Jacob Lurie recovered this approach and considerably strengthened it in his paper [3]: by now we define an "n-dimensional, k-valued Topological Quantum Field Theory"as a monoidal functor between the category of n-cobordisms and that of finite dimensional k-vector spaces.

If you have time and spirit to do so, I warmly recommend to give a glance to a "simplified n-categorical formulation" of TQFTs given by B. Toen in his [4].

References.

[1] Atiyah, Michael (1988), "Topological quantum field theories", Publications Mathématiques de l'IHÉS 68 (68): 175–186

[2] Samuel Eilenberg, Norman E. Steenrod, Axiomatic approach to homology theory, Proc. Nat. Acad. Sci. U. S. A. 31, (1945). 117–120.

[3] Lurie, Jacob, On the Classification of Topological Field Theories

[4] Bertrand Toen, Higher categorical structures in TQFT. Lecture in MPI-Bonn, June 2000.

• I haven't looked at [1],[3],and [4] recently, and maybe some people call the construction described in the question a TQFT nowadays (I'm not entirely convinced, though). As a non-topologist/non-mathematical-physicist I tend to agree with Ryan's comment: It's a cobordism category.
– t.b.
Commented May 16, 2011 at 6:52
• I am also aware of this line. I would like to hear more from you about the intuition behind this construction. Commented May 17, 2011 at 7:43