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An exciting breakthrough has occurred in the field of topology: a group of mathematicians has falsified the "telescope conjecture". This conjecture was proposed by Douglas Lavernier in 1984 and is best known as the last of his series of conjectures to be solved.
The team announced their findings at the "Panorama of Homotopy Theory" conference organized by the Isaac Newton Institute for Mathematical Sciences (INI) in Cambridge, where Tomer Schlank, Jeremy Hahn, Robert Burklund, and Ishan Levy demonstrated the results. We sat down with Tomer Schranke and Jeremy Hahn to talk about their work in the simplest language possible. Here's what they told us.
What is Topology?
When it comes to geometry, topology takes a relaxed attitude as we do in our daily lives. When we say that a golf ball is spherical, we don't care that it has many small pits, which technically means that it is not a perfect sphere. Similarly, despite the obvious bumpiness of oranges and apples, we would also consider them to be round.
Topology shows this tolerance for deformation. If two shapes can be converted into each other by extrusion or stretching (rather than cutting or gluing), then they are considered topologically identical. In this sense, golf balls, oranges, apples and leaky soccer balls are all the same. Similarly, donuts are topologically identical to coffee cups (as shown in the GIF below). However, a doughnut is not the same as a spherical surface, because the only way to turn a spherical doughnut into a doughnut is to cut a hole in the spherical surface and glue the edges together.
The difference between a donut (strictly a torus) and a sphere is that a torus has a hole, while a sphere does not. In fact, holes are extremely important in topology. Mathematicians have shown that many of the surface structures that come to mind naturally are topological equivalent to a sphere (i.e., without any holes), a torus (with one hole), or a torus with two or three holes, etc. In topology, we define a surface using the number of holes.
Take a loop
Now you may be asking, "What is a hole?" and when we see a torus, it's clear that it does have a hole. But once you try to put the "holed" property into words, things get a little tricky.
At this point, we can use circles to help. Any circle you might draw on the topology ball can be contracted to a single point. For topological torus, or other surfaces with holes, the situation is a bit different: if the circle rotates around the hole, you can push or move it as much as you want, but as long as you don't glue the surfaces together, it will never shrink to a single point.
If your topological ball is a golf ball, or your topological torus is a curved doughnut, you can't draw a perfect circle on either side due to the depressions and bumps. But we don't have to worry about that, we're looking for a general circle, a closed loop.
Again, we don't have to distinguish between loops that do not cut, but can be converted into each other by, for example, sliding, stretching or extruding on the surface. These loops can be thought of as equivalent, and we can divide the set of loops drawn on the surface into equivalence classes. If two loops can be converted to each other without cutting, they are in the same class, and in the image above, loops of the same color belong to the same equivalent class.
All of the above can be mathematically and precisely: the equivalence classes of the apparent loop are called homotopy classes, and the set of all these equivalence classes is called the homotopy group of the surface. The exact composition of the homotopy group provides information about the holes in the surface and is therefore able to tell us a lot about the surface itself (yes, it constitutes a group mathematically).
Two-dimensional holes
Once you recognize the importance of holes in describing surfaces, and the importance of loops in describing holes, you may be asking yourself how you can proceed.
So far, we've considered all the possible ways to be able to draw a loop on a surface, in other words, to be able to map a circle continuously on a surface. "Mapped" a circle simply means assigning each point on the circle to a point on the surface, while "continuous" means that there are no gaps in the resulting loop.
A circle is a one-dimensional analogy of a sphere. If we take it one more dimension, what other way can we map the ball to our surface continuously? If the surface itself is a topological ball, let's say a leaky soccer ball, then it's easy to imagine a way to do that. You can deform the sphere until it has the exact same shape as the soccer ball, and then assign each point on the deformed sphere to the corresponding point on the soccer ball.
You can also easily map a sphere to a shape that contains a topology sphere, as shown in the image below.
However, you'll find that you can't map the sphere to the topological torus, and the holes become a problem again.
In the first two examples, once the sphere is mapped onto a surface, it is no longer possible to shrink the sphere to a point without the surface itself being deformed. This is because it encloses what you think of as a two-dimensional hole, which is the interior of the topological ball you want to map to.
"The sphere doesn't have any one-dimensional holes, because every time you draw a loop, you can shrink it," says Tomer Schrank. "But the sphere does have a two-dimensional hole, because you can map another sphere onto it so that it can't be shrunk. This is not intuitive, but it is not difficult to prove that there are no two-dimensional holes in the torus. ”
These ideas can be expressed precisely in the language of mathematics. Similar to what we saw earlier about loops, you can end up with a homotopy group of a surface that tells you about the presence of a two-dimensional hole. In general, this homotopy group is generated by all methods of continuously mapping the sphere to other surfaces, and can be used to determine whether the mapped surface can shrink to a point.
High-dimensional holes and shapes
Now, we know that a circle is a one-dimensional object, while a sphere is a plane, i.e., two-dimensional. Although we can't visualize high-dimensional surfaces, mathematicians have ways of defining them. So you can work with them in the same way as you would with circles or normal spheres.
Similarly, you can now define homotopy groups that map high-dimensional spheres to a given shape. These homotopy groups will provide you with information about what you think are high-dimensional holes in the shape.
"There is something surprising and unanticipated in the definition of homotopy groups, and that is that you can get high-dimensional holes in low-dimensional spheres," Schranke explains. "For example, you can write down a continuous map from a 3D sphere to a 2D sphere and not shrink to a single point. "This means that this two-dimensional sphere has a three-dimensional hole. Although it is beyond what we can imagine, the image below depicts this mapping. You don't need to understand it – we're putting it here just to say that mathematicians do have a clear idea of what they're talking about.
Now, we can consider whether a 2D sphere has holes in higher dimensions such as 4D, 5D, 6D, etc. More generally, after giving the dimension n of the sphere, it is possible to consider whether there is a hole with a higher dimension m. The corresponding homotopy group is denoted by the term. The subscript m is the dimension of the hole to be looked for, while the superscript n is the dimension of the surface on which the hole is located.
"Understanding the general question of 'what is the homotopy group generated by the mapping between spheres' remains a key topic in the field of homotopy theory," Schranke said. In other words, mathematicians hope to be able to further understand homotopy groups that traverse all combinations of m and n.
This has proven to be an impossible task at the moment, but the good thing is that the nature of mathematics can provide us with effective simplification. The mathematician Hans Freudenthal (1937) proved that homotopy groups are the same as long as the difference in the contained dimensions (m-n) is the same and n is large enough (strictly speaking, they are isomorphic to each other). This means that there are homotopy groups that are identical to each other throughout the branch. For example:
Here, the difference in dimensionality is 1.
Here, the difference in dimensionality is 2.
Here, the difference in dimensionality is 3.
The symbol represents that these groups are essentially the same (i.e., isomorphic to each other).
The homotopy groups in these branches are called stable homotopy groups. For each integer, there is a stable homotopy group: it contains a homotopy group with a dimension difference of 1, a dimensionally group with a dimension difference of 2, and so on. So we can start with a relatively simple problem and look at these stable homotopy groups, not all of them. Schranke said.
Dramatic failure
Stabilizing homotopy groups makes the problem easier, but it's not easy to understand. Douglas Lavinier proposed the telescope conjecture in 1984. He has said that he would not expect his grandchildren to fully understand everything in their lifetimes. This is why mathematicians have stopped focusing on individual stable homotopy groups and have instead tried to understand their overall structure. Lavenier likens this behavior to being in a huge mansion. Rather than investigating every room, you prefer to understand the structure of the building as a whole.
Mathematician Hans Freidenthal made a great contribution to homotopy theory. The portrait was taken in 1957. Photo: Hofland, L.H., Het Utrechts Archief, CC BY 4.0.
The telescope conjecture makes it possible to grasp the structure of the homotopy group as a whole. But now, Berklund, Hahn, Levy and Schranke have proved that conjecture wrong. This means that the method offered by Lavenier will not work. "I think it's fair to say that we've not only proven that this approach doesn't work, but that it failed very completely. The mansion was much more complicated than we expected. Hahn said.
Hope remains
But that doesn't mean we're back to square one. Our knowledge of stable homotopy groups suggests that there is some pattern in their ensemble. You can break this pattern down into light-like wavelengths, with each individual wavelength exhibiting periodicity. "These cycles do exist, but they are far more complex than the telescope conjecture might expect," Schranke said. "However, the way we falsified the telescope conjecture has given us some useful insights," he added. "For example, it provides a lower bound for the size of the homotopy group. ”
Bercklund, Hahn, Levy and Schlanke announced their results at the Oxford conference to great acclaim. "This is a special moment for all of us," says Hahn. "Lavenier was there, and so was Mike Hopkins. Together with his collaborators, he proved the vast majority of Lavenier's conjectures, with the exception of the telescope conjecture. The entire history of people who have crossed paths with these conjectures gathered in the conference room. We are very grateful for this conference, which brought all these people together. Now seems like the perfect time to explain our work. In fact, the meeting was a tribute to Mike Hopkins. This is part of a two-week event organized by the Isaac Newton Institute for Mathematical Sciences (INI). The second part, "Tonglun: The Fruit of Fertile Land", was held at the INI in Cambridge.
After nearly 40 years, after Lavenier's conjecture was fully resolved, the field of homotopy theory began to move towards a new direction of development to better understand stable homotopy groups. INI has played its part in the exploration of this area: the recent conference is a continuation of a research program launched in 2018. Perhaps in the future, it will also host the publication of important research results. If we were still present, we would definitely report on this information.
作者:Marianne Freiberger
Translation: wnkwef
Reviewer: Yue Yue
The views expressed in the translation are solely those of the author
It does not represent the position of the Institute of Physics of the Chinese Academy of Sciences
Edit: There is a lot of interest