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cobordism.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1//EN"
"http://www.w3.org/TR/xhtml11/DTD/xhtml11.dtd">
<html xmlns="//www.w3.org/1999/xhtml" xml:lang="en">
<head>
<meta name="generator" content="jemdoc, see http://jemdoc.jaboc.net/" />
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<link rel="stylesheet" href="jemdoc.css" type="text/css" />
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<script src='update_info.js'></script>
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document.getElementById('footer-text').innerHTML = 'This page last updated ' +
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<script src='cobordism.js'></script>
<title>Unoriented cobordism</title>
</head>
<body>
<h1>Cobordism groups</h1>
<div id="layout-content">
<h2>Unoriented cobordism</h2>
<p>
René Thom's 1954 paper <a href='http://www.maths.ed.ac.uk/~aar/papers/thomcob.pdf'><i>Quelques propriétés globales
des variétés differentiables</i></a> calculated the unoriented <a
href='https://en.wikipedia.org/wiki/Cobordism'>cobordism ring</a> as a free polynomial algebra over 𝔽<sub>2</sub>
with a single generator in each degree <i>k</i> such that <i>k</i> + 1 is not a power of 2. Hence the size of the
cobordism group in dimension <i>n</i> is a modified partition function. This calculator computes this function
given <i>n</i>.
</p>
<form name='MO'>
<input type='text' id='input_MO' onKeyDown="if (event.keyCode == 13) {main('MO'); return false}">
<input type='button' value='Go' onClick='main("MO")'>
</form>
<br>
<p>
Answer: dim<sub>𝔽<sub>2</sub></sub> Ω<sub><span id='dim_MO'><i>n</i></span></sub> = <span id='answer_MO'>_</span>.
</p>
<!-- next: cobordisms with a principal Z/2-bundle -->
<h2>Unoriented manifolds with a principal ℤ/2-bundle</h2>
<p>
Under cobordism, closed, unoriented manifolds with a principal ℤ/2-bundle form a ring. Thom's work identifies this
group with π<sub>*</sub>(<i>MO</i> ∧ <i>B</i>ℤ/2<sub>+</sub>), which can be computed in a similar manner as above.
</p>
<form name='BZ2'>
<input type='text' id='input_BZ2' onKeyDown="if (event.keyCode == 13) {main('BZ2'); return false}">
<input type='button' value='Go' onClick='main("BZ2")'>
</form>
<br>
<p>
Answer: dim<sub>𝔽<sub>2</sub></sub> π<sub><span id='dim_BZ2'><i>n</i></span></sub>(<i>MO</i> ∧
<i>B</i>ℤ/2<sub>+</sub>) = <span id='answer_BZ2'>_</span>.
</p>
<h2>Complex cobordism</h2>
<p>
Novikov's 1960 paper <i>Some problems in the topology of manifolds connected with the theory of Thom spaces</i>
uses Thom's construction to calculate that the cobordism ring of stably almost complex manifolds is a free
polynomial algebra over ℤ. Again, the size of the cobordism group in dimension <i>n</i> is a modified partition
function, computed in a similar way.
</p>
<form name='MU'>
<input type='text' id='input_MU' onKeyDown="if (event.keyCode == 13) {main('MU'); return false}">
<input type='button' value='Go' onClick='main("MU")'>
</form>
<br>
<p>
Answer: dim<sub>ℤ</sub> π<sub><span id='dim_MU'><i>n</i></span></sub>(<i>MU</i>) = <span id='answer_MU'>_</span>.
</p>
<footer>
<div id="footer-text"></div>
</footer>
</div>
</body>
</html>