{"id":489,"date":"2012-11-28T18:18:47","date_gmt":"2012-11-28T17:18:47","guid":{"rendered":"http:\/\/boolesrings.org\/matsguru\/?p=55"},"modified":"2012-11-28T18:18:47","modified_gmt":"2012-11-28T17:18:47","slug":"the-significance-of-adipra","status":"publish","type":"post","link":"https:\/\/matsguru.com\/?p=489","title":{"rendered":"The significance of adipra"},"content":{"rendered":"

I realize that anyone reading this post doesn’t know what adipra is. How can you? I made it up.<\/p>\n

The sphere I constructed most recently was an idea of Karim Adiprasito<\/a>. While building it, I needed to name it so that I can refer to it in my code. So I temporarily named it adipra and that’s what I’m going to call it here.<\/p>\n

Actually, in this post, I don’t want to talk about what adipra is. Not yet. Let’s talk just about what makes adipra special. For anyone who’s interested, all the specifics can be found here<\/a>.<\/p>\n

We need a little language to get started.<\/p>\n

Def<\/span><\/strong> A combinatorial <\/em>d-manifold<\/em> is a triangulated d<\/em>-manifold whose vertex links are PL spheres.<\/p>\n

Let’s assume we know what a\u00a0triangulated d<\/em>-manifold means without hashing out the details here. If you need to, you can think of it as a simplicial complex.<\/p>\n

The star<\/em> of a vertex v<\/em> in a triangulated d<\/em>-manifold T<\/em> is the collection of facets of T<\/em> that contain v<\/em>. The link<\/em> of a vertex v<\/em> is like the star of v<\/em>, but take out all the v<\/em>‘s. For example, let’s take a simple hexagon with a vertex in the center.<\/p>\n

\"vertexlink\"
Example of star and link of vertex<\/figcaption><\/figure>\n

Let T<\/em>=$\\{[0\\, 1 \\,2],[0\\, 1\\, 6],[0\\, 2\\, 3],[0\\, 3\\, 4],[0\\, 4\\, 5],[0\\, 5\\, 6]\\}$, then \\begin{align*}\\text{star}(3,T) &=\\{[0\\, 2 \\,3],[0\\, 3\\, 4]\\} \\text{ (green triangles)}\\\\ \\text{link}(3,T)&=\\{[0\\, 2],[0\\, 4]\\} \\text{ (red edges)} \\end{align*}<\/p>\n

That’s simple enough to understand even in higher dimensions. Just remember we’re always doing things combinatorially.<\/p>\n

Next is the PL sphere. I said earlier that we can assume we know what a triangulated d<\/em>-manifold is. What we’re actually talking about is a triangulated PL d<\/em>-manifold. A PL-sphere<\/em> is a PL manifold that is bistellarly equivalent to the boundary of a d<\/em>-simplex. I’ll write in more detail what bistellarly equivalent means in a later post. For now, just think of it as the discrete version of being homeomorphic.<\/p>\n

Putting all that together, we understand a combinatorial d<\/em>-manifold to be a triangulated manifold, that is, some simplicial complex-like thing where we require that each of its verticies is sort of covered by a ball. Naturally, the next question to ask is: are there triangulated manifolds that are not combinatorial?<\/p>\n

For d<\/em>=2,3, all triangulations (of the d<\/em>-sphere) are combinatorial. For d<\/em>=2, the vertex links should be homeomorphic to $S^1$ or bistellarly equivalent to the triangle (boundary of a 2-simplex). Similarly, for d<\/em>=3, vertex links are $S^2$. For d<\/em>=4, all triangulated 4-manifolds are also combinatorial. This result is due to Perelman. The vertex links are 3-spheres, which, as you know, is what Perelman worked on. [Remember that PL=DIFF in dim 4.] But it falls apart for d<\/em>$\\ge 5$ as there are non-PL triangulations of the d<\/em>-sphere.<\/p>\n

Ok, so here’s a spoiler about adipra: it’s a non-PL triangulation of the 5-sphere. But that’s not all.<\/p>\n

There’s a nice theorem by Robin Forman, the father of discrete Morse theory.<\/p>\n

Theorem<\/span><\/strong> (Forman) Every combinatorial d<\/em>-manifold that admits a discrete Morse function with exactly two critical cells is a combinatorial d<\/em>-sphere.<\/p>\n

Actually, this is not really Forman’s theorem. This is a theorem by Whitehead which Forman reformulated using his language of discrete Morse theory. This is the original theorem.<\/strong><\/p>\n

Theorem<\/span><\/strong> (Whitehead) Any collapsible combinatorial d<\/em>-manifold is a combinatorial d<\/em>-ball.<\/p>\n

What Forman did is take a sphere, take out one cell, call that guy a critical cell, then collapse the rest of it down (using Whitehead’s theorem) to get the other critical cell. Thus you have two critical cells.<\/p>\n

So the question you ask next is: can you have non-combinatorial spheres with 2 critical cells?<\/p>\n

Yes, actually, you can! Karim showed that you can have a non-PL\/non-combinatorial triangulation of the 5-sphere that has a discrete Morse function with exactly 2 critical cells! And then I built an explicit example (with Karim’s instructions) and called it adipra.<\/p>\n","protected":false},"excerpt":{"rendered":"

I realize that anyone reading this post doesn’t know what adipra is. How can you? I made it up. The sphere I constructed most recently was an idea of Karim Adiprasito. While building it, I needed to name it so that I can refer to it in my code. So I temporarily named it adipra…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[14],"tags":[16,17],"class_list":["post-489","post","type-post","status-publish","format-standard","hentry","category-spheres","tag-adipra","tag-adiprasito"],"_links":{"self":[{"href":"https:\/\/matsguru.com\/index.php?rest_route=\/wp\/v2\/posts\/489","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/matsguru.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/matsguru.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/matsguru.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/matsguru.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=489"}],"version-history":[{"count":0,"href":"https:\/\/matsguru.com\/index.php?rest_route=\/wp\/v2\/posts\/489\/revisions"}],"wp:attachment":[{"href":"https:\/\/matsguru.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=489"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/matsguru.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=489"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/matsguru.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=489"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}