|
|
| Line 1: |
Line 1: |
| In mathematics, the Toda bracket is an operation on homotopy classes of maps, in particular on [[homotopy groups of spheres]], named after [[Hiroshi Toda]] who defined them and used them to compute homotopy groups of spheres in {{harv|Toda|1962}}.
| | Jayson Berryhill is how I'm known as and my wife doesn't like it at all. I've usually cherished living in Alaska. To play lacross is the factor I adore most of all. Distributing manufacturing is exactly where her main income arrives from.<br><br>Review my webpage - online psychics ([http://www.edmposts.com/build-a-beautiful-organic-garden-using-these-ideas/ www.edmposts.com]) |
| | |
| == Definition ==
| |
| See {{harv|Kochman|1990}} or {{harv|Toda|1962}} for more information.
| |
| Suppose that
| |
| :<math>W\stackrel{f}{\ \to\ } X\stackrel{g}{\ \to\ } Y\stackrel{h}{\ \to\ } Z</math>
| |
| is a sequence of maps between space, such that ''gf'' and ''hg'' are both nullhomotopic. Given a space ''A'', let ''CA'' denote the cone of ''A''. Then we get a non-unique map from ''CW'' to ''Y'' from a homotopy from ''gf'' to a trivial map, which when composed with ''h'' gives a map from ''CW'' to ''Z''. Similarly we get a non-unique map from ''CX'' to ''Z'' from a homotopy from ''hg'' to a trivial map, which when composed with ''Cf'', the cone of the map ''f'', gives another map from ''CW'' to ''Z''. By joining together these two cones on ''W'' and the maps from them to ''Z'', we get a map 〈''f'', ''g'', ''h''〉 in the group [''SW'', ''Z''] of homotopy classes of maps from the suspension SW to ''Z'', called the '''Toda bracket''' of ''f'', ''g'', and ''h''. It is not uniquely defined up to homotopy, because there was some choice in choosing the maps from the cones. Changing these maps changes the Toda bracket by adding elements of ''h''[''SW'',''Y''] and [''SX'',''Z'']''f''.
| |
| | |
| There are also higher Toda brackets of several elements, defined when suitable lower Toda brackets vanish. This parallels the theory of [[Massey product]]s in [[cohomology]].
| |
| | |
| == The Toda bracket for stable homotopy groups of spheres ==
| |
| The [[Direct sum of groups|direct sum]]
| |
| :<math>\pi_{\ast}^S=\bigoplus_{k\ge 0}\pi_k^S</math>
| |
| of the stable homotopy groups of spheres is a [[supercommutative ring|supercommutative]] graded [[ring (mathematics)|ring]], where multiplication (called composition product) is given by composition of representing maps, and any element of non-zero degree is [[nilpotent]] {{harv|Nishida|1973}}.
| |
| | |
| If ''f'' and ''g'' and ''h'' are elements of π<sub>∗</sub><sup>''S''</sup> with ''f'' ⋅ ''g'' = 0 and ''g'' ⋅ ''h'' = 0, there is a ''Toda bracket'' 〈''f'', ''g'', ''h''〉 of these elements. The Toda bracket is not quite an element of a stable homotopy group, because it is only defined up to addition of composition products of certain other elements. [[Hiroshi Toda]] used the composition product and Toda brackets to label many of the elements of homotopy groups.
| |
| {{harvtxt|Cohen|1968}} showed that every element of the stable homotopy groups of spheres can be expressed using composition products and higher Toda brackets in terms of certain well known elements, called Hopf elements.
| |
| | |
| == References ==
| |
| *{{citation
| |
| |last= Cohen|first= Joel M.
| |
| |title= The decomposition of stable homotopy.
| |
| |journal= [[Annals of Mathematics]] (2)
| |
| |volume= 87
| |
| |year= 1968
| |
| |pages= 305–320
| |
| |doi=10.2307/1970586
| |
| |mr= 0231377
| |
| |issue= 2
| |
| |jstor=1970586}}.
| |
| * {{citation
| |
| |last= Kochman
| |
| |first= Stanley O.
| |
| |chapter=Toda brackets
| |
| |pages=12–34
| |
| |title= Stable homotopy groups of spheres. A computer-assisted approach
| |
| |series= Lecture Notes in Mathematics
| |
| |volume= 1423
| |
| |publisher= [[Springer-Verlag]]
| |
| |publication-place =Berlin
| |
| |year= 1990
| |
| |isbn= 978-3-540-52468-7
| |
| |mr= 1052407
| |
| |doi=10.1007/BFb0083797
| |
| }}.
| |
| * {{citation
| |
| |last= Nishida | first= Goro | authorlink = Goro Nishida
| |
| |title= The nilpotency of elements of the stable homotopy groups of spheres
| |
| |journal=Journal of the Mathematical Society of Japan
| |
| |volume= 25
| |
| |year= 1973
| |
| |pages= 707–732
| |
| |issn= 0025-5645
| |
| |mr= 0341485
| |
| |doi= 10.2969/jmsj/02540707
| |
| |issue= 4
| |
| }}.
| |
| * {{citation
| |
| |last= Toda
| |
| |first= Hirosi
| |
| |title= Composition methods in homotopy groups of spheres
| |
| |publisher= [[Princeton University Press]]
| |
| |year= 1962
| |
| |isbn= 978-0-691-09586-8
| |
| |mr=0143217
| |
| |series=Annals of Mathematics Studies
| |
| |volume=49
| |
| }}.
| |
| | |
| [[Category:Homotopy theory]]
| |
Jayson Berryhill is how I'm known as and my wife doesn't like it at all. I've usually cherished living in Alaska. To play lacross is the factor I adore most of all. Distributing manufacturing is exactly where her main income arrives from.
Review my webpage - online psychics (www.edmposts.com)