# What exactly did Poincaré mean by 'simply connected'?

I've been reading John Stillwell's translation of the famous Analysis Situs and have become confused about the exact meaning of 'simply connected' in Poincaré's language. On page 7 (in the introduction), Stillwell claims Poincaré defines it, in paragraph 14, as a manifold with trivial fundamental group (as is the modern sense of the term). However, in the actual text I find no such definition clearly stated. The term 'simply connected' is first used on page 65, seemingly with no definition given. On page 74 we have something suggestive:

Thus we have three manifolds whose group are of finite order, but non-isomorphic, so the manifolds cannot be homeomorphic. Nevertheless, they have the same Betti numbers $$P_1 = P_2 = 1.$$ It seems natural to restrict the meaning of the term simply connected to manifolds whose group $G$ reduces to a single substitution. Then a closed manifold of more than two dimensions can have a group $G$ of finite order without being simply connected.

I can only interpret this as meaning that, to Poincaré, simply connected meant trivial fundamental group plus something else. It seems to me like this something else is all Betti numbers being equal to 1 (note that Poincaré's Betti numbers differ from the modern ones, being shifted up by 1).

Furthermore, on the very last page of the fifth (and last) complement, Poincaré states his famous conjecture:

Is it possible for the fundamental group of $V$ to reduce to the identity without $V$ being simply connected?

(Poincaré goes on to end the paper claiming, ironically enough, "However, this question would carry us too far away.") This would seem like a silly question if simply connected were a synonym of trivial fundamental group (unless I misunderstood what he means by 'reduce to the identity' -- I am interpreting this as being a trivial group).

Even if understood as I suggested above, this is still a bit strange a question, as it is vastly different from what gets called the Poincaré conjecture nowadays -- in fact, it's easy to show that a simply connected (in the modern understanding of the term) closed 3-manifold is a homology sphere (in particular, has the same Betti numbers as the sphere), and so is simply connected even in the more restrictive definition. Indeed, the proof is just Poincaré duality, which obviously Poincaré knew about, so he couldn't have missed it. Nevertheless, on a footnote Stillwell claims this is a correct statement of Poincaré's conjecture.

How to make light of this confusion? What did Poincaré mean by 'simply connected' and how did he interpret his own conjecture?

In the setting of the conjecture (closed manifolds) he certainly meant “homeomorphic to the $n$-sphere” — see second page of that fifth complement:

simplement connexe au sens propre du mot, c'est-à-dire homéomorphe à l'hypersphère.

(Stillwell, p. 179. Same on pp. 141, 152, 169, 255. Stillwell apparently missed the subtlety you note that what is said on p. 74 is not a definition, so he calls Poincaré confused in his Introduction, p. 7.)

Edit:
In general Poincaré’s meaning was “homeomorphic to a ball or a sphere”, according to H. P. de Saint-Gervais’ site Analysis Situs, or this book — which however don’t seem to source the claim.

Poincare really was confused in the beginning. He probably wanted "simply connected" to mean "homeomorphic to the sphere", as F. Ziegler points in his answer, and tried to find a condition of this in terms of the homology groups and the fundamental group which he introduced. In the beginning he thought that one can characterize the sphere in terms of homology groups, and even had a wrong "proof" of this. Then he discovered what is now called "homology spheres" or "Poincare spheres" which have the same homology as the spheres but not homeomorphic to them. Whether the trivial fundamental group implies the sphere is the celebrated Poincare conjecture, now proved.

When you read Poincare, take into account that he lived long before Bourbaki, and his terms do not always have exact meaning. Or this meaning could change as his knowledge expanded.

• Then do you know what he could mean by 'simply connected' when talking about manifolds with boundary? e.g. in his definition of polyhedron, he assumes cells and faces to be simply connected. – Pedro Jun 15 '17 at 14:49
• Homeomorphic to the sphere. – Alexandre Eremenko Jun 16 '17 at 5:23