Poincaré Conjecture and the Shape of the Universe
In Einstein's theory of General Relativity, the universe is a 4-manifold that might well be fibered by 3-dimensional time slices. If a particular spacetime that doesn't have such a fibration, then it is difficult to construct a causal model of the laws of physics within it. (Even if you don't see an a priori argument for causality, without it, it is difficult to construct enough solutions to make meaningful predictions.) There isn't usually a geometrically distinguished fibration, but if you have enough symmetry or even local symmetry, the symmetry can select one. An approximate symmetry can also be enough for an approximately canonical fibration. Once you have all of that, the topology of spacelike slices of the universe is not at all a naive or risible question, at least not until you see more physics that might demote the question. The narrower question of whether the Poincaré Conjecture is relevant is more wishful and you could call it naive, but let's take the question of relating 3-manifold topology in general to cosmology.
The cosmic microwave background, discovered in the 1964 by Penzias and Wilson, shows that the universe is very nearly isotropic at our location. (The deviation is of order $10^{-5}$ and it was only announced in 1992 after 2 years of data from the COBE telescope.) If you accept the Copernican principle that Earth isn't at a special point in space, it means that there is an approximately canonical fibration by time slices, and that the universe, at least approximately and locally, has one of the three isotropic Thurston geometries, $E^3$, $S^3$, or $H^3$. The Penzias-Wilson result makes it a really good question to ask whether the universe is a 3-manifold with some isotropic geometry and some fundamental group. I have heard of the early discussion of this question was so naive that some astronomers only talked about a 3-torus. They figured that if there were other choices from topology, they could think about them later. Notice that already, the Poincaré conjecture would have been more relevant to cosmology if it had been false!
The topologist who has done the most work on the question is Jeff Weeks. He coauthored a respected paper in cosmology and wrote an interesting article in the AMS Notices that promoted the Poincaré dodecahedral space as a possible topology for the universe. But after he wrote that article...
There indeed is other physics that does demote the 3-manifold question, and that is inflationary cosmology. The inflation theory posits that the truthful quantum field theory has a vaguely stable high-energy phase, which has such high energy density that the solution to the GR equations looks completely different. In the inflationary solution, hot regions of the universe expand by a factor of $e$ in something like $10^{-36}$ seconds. The different variations of the model posit anywhere from 60 to thousands of factors of $e$, or "$e$-folds". Patches of the hot universe also cool down, including the one that we live in. In fact every spot is constantly cooling down, but cooling is still overwhelmed by expansion. Instead of tacitly accepting certain observed features of the visible universe, for instance that it is approximately isotropic, inflation explains them. It also predicts that the visible universe is approximately flat and non-repeating, because macroscopic curvature and topology have been stretched into oblivion, and that observable anisotropies are stretch marks from the expansion. The stretch marks would have certain characteristic statistics in order to fit inflation. On the other hand, in the inflationary hot soup that we would never see directly, the rationale for canonical time slices is gone, and the universe would be some 4-manifold or even some fractal or quantum generalization of a 4-manifold.
The number of $e$-folds is not known and even the inflaton field (the sector of quantum field theory that governed inflation) is not known, but most or all models of inflation predict the same basic features. And the news from the successor to COBE, called WMAP, is that the visible universe is flat to 2% or so, and the anistropy statistically matches stretch marks. There is not enough to distinguish most of the models of inflation. There is not enough to establish inflation in the same sense that the germ theory of disease or the the heliocentric theory are established. What is true is that inflation has made experimental predictions that have been confirmed.
After all that news, the old idea that the universe is a visibly periodic 3-manifold is considered a long shot. WMAP didn't see any obvious periodicity, even though Weeks et al were optimistic based on its first year of data. But I was told by a cosmologist that periodicity should still be taken seriously as an alternative cosmological model, if possibly as a devil's advocate. A theory is incomplete science if it is both hard to prove, and if every alternative is laughed out of the room. In arguing for inflation, cosmologists would also like to have something to argue against. In the opinion of the cosmologist that I talked to some years ago, the model of a 3-manifold with a fundamental group, developed by Weeks et al, is as good at that as any proposal.
José makes the important point that, in testing whether the universe has a visible fundamental group, you wouldn't necessarily look for direct periodicity represented by non-contractible geodesics. Instead, you could use harmonic analysis, using a suitable available Laplace operator, and this is what used by Luminet, Weeks, Riazuelo, Lehoucq and Uzan. I also that I have not heard of any direct use of homotopy of paths in astronomy, but actually the direct geometry of geodesics does sometimes play an important role. For instance, look closely at this photograph of galaxy cluster Abell 1689. You can see that there is a strong gravitational lens just left of the center, between the telescope and the dimmer, slivered galaxies. Maybe no analysis of the cosmic microwave background would be geometry-only, but geometry would modify the apparent texture of the background, and I think that that is part of the argument from the data that the visible universe is approximately flat. Who is to say whether a hypothetical periodicity would be seen with geodesics, harmonic expansion, or in some other way.
Part of Gromov's point seems fair. I think it is true that you can always expand the scale of proposed periodicity to say that you haven't yet seen it, or that the data only just starts to show it. Before they saw anisotropy with COBE, that kept getting pushed back too. The deeper problem is that the 3-manifold topology of the universe does not address as many issues in cosmology, either theoretical or experimental, as inflation theory does.
No, I believe it has not and that it is hard to see how it could be directly useful.
Just for starters, it is not clear whether the universe is simply connected. It is not even clear that this is a meaningful question. In this regard I cannot resist quoting a passage from Steven Krantz' Mathematical Apocrypha Redux:
Mikhael Gromov tells of attending lectures about cosmology by two topologists -- really great topologists -- concerning the possible shape of the Universe. Gromov asked the first of these whether the Universe was simply connected. The man replied, "It is clear that the Universe cannot be but simply connected, for non-simple connectedness would imply some high-scale periodicity, which is ridiculous." The other topologist gave a talk entitled "Is the Universe simply connected?" which seemed to be related to the question. When informed of the first speaker's statement, he said, "Who cares, it's still a meaningful question, like it or not." In discussing these divergent opinions, Gromov offers his own: "Take a loop in the Universe, a reasonably short loop compared to the size of the Universe, say of no more than $10^{10}$ to $10^{12}$ light years long and ask if it is contractible. And, to be realistic, we pick a certain time, for example $10^{30}$ years, and ask if it is contractible within this time. So you are allowed to move the loop around, say at the speed of light, and try to determine whether or not it can be contracted within this time. The point is, even imagining our space to be some topological 3-sphere S3, we can organize an innocuous enough metric on S3 so that it takes more than $10^{30}$ years to contract certain loops in this sphere and in the course of contraction we need to stretch the loop to something like $10^{30}$ light years in size. So, if $10^{30}$ years is all the time you have, you conclude that the loop is not contractible and whether or not $\pi_1(S^3) = 0$ becomes a matter of opinion."
On the other hand, the issue of applicability of Poincare (and geometrization) to the real world has been raised by others. The government agency DARPA listed the following as one of its 23 Challenges for the Future:
What are the Physical Consequences of Perelman's Proof of Thurston's Geometrization Theorem? Can profound theoretical advances in understanding three dimensions be applied to construct and manipulate structures across scales to fabricate novel materials?
To me DARPA's question seems naive bordering on risible, but others may feel differently, especially if they receive funding.
With all due respect to Mikhail Gromov, Physics is not about homotoping loops in the spatial universe! That would certainly be risible. In Physics, the fundamental group reflects itself in the spectrum of the laplacian: $S^3$ and $S^3/\Gamma$, say, have different harmonics and this is something that can be infered from measurements, even if perhaps not measured directly.
According to current cosmological models, the early universe was a hot plasma which light could not escape: basically photons had a very small mean free path because there is so much activity in the plasma that they find something to scatter against before long. It is only after the universe starts cooling that at some point the photons can escape. This happens over a period of time, but cosmologically speaking we can assume it happens instantaneously. The cosmic microwave background we measure today are the photons which were released at that time. They have of course cooled down considerably since then. They contain information about the "surface of last scatter" and what experiments such as COBE, WMAP, Planck,... are essentially doing is taking pictures of the surface of last scatter with increasing resolution. The colours in the pretty pictures you see coming from COBE and WMAP correspond to temperature fluctuations which can be mapped to density fluctuations in the early universe via something called the Sachs-Wolfe effect. In a nutshell, this effect relates the harmonic expansion of the density fluctuations in the early spatial universe and the harmonic expansion of the temperature as a function on the celestial sphere. The latter can be measured directly, and the former can be inferred. So in principle it is possible to relate the topology of the spatial universe to empirical data.
In fact the work of Luminet, Weeks, Riazuelo, Lehoucq and Uzan that Greg Kuperberg links to in his answer was based on the analysis of the first-year data from WMAP. The power spectrum of the data revealed that the lowest lying modes were severely attenuated and this suggested that the spatial universe was missing the lowest lying harmonics: recall that the harmonics in $X/\Gamma$ are the $\Gamma$-invariant harmonics on $X$. Of the possible models, it was $S^3/E_8$ which fit the (very scant!) data best. This model has now been deprecated because it predicts the existence of "circles in the sky": sectors of circles in two different locations of the celestial sphere with the same temperature fluctuations. There are computer algorithms that search for those circles in the sky but as far as I know none were ever found.
This story suggests the question: Can one hear the shape of the universe?, paraphrasing Mark Kac's famous question about the shape of a drum. In the case of Kac's question we know the answer to be negative: Milnor found that there are non-isometric isospectral 16-dimensional tori, coincidentally the two tori which define the two heterotic string theories, and more recently also non-isometric planar domains with piecewise linear boundaries have been constructed which are isospectral. However for three-dimensional space forms the answer to the question is positive, and this is gives hope that cosmological data might determine the shape of universe.
Whether the Poincaré conjecture has any bearing on this story is not clear to me, but what is undeniable is that the nature of the topology and the geometry of the spatial universe is physically relevant. It bears upon such "big" questions as whether the universe is finite or infinite and indeed as to what its ultimate fate might be, whether or not Gromov or indeed any of us, will be there to witness it.