Different Categories
The above bijection gives a structure of an abelian group since the space is a loop space and in fact an infinite loop space so the normal invariants are a zeroth cohomology group of an extraordinary cohomology theory defined by that inifinite loop space. Note that similar ideas apply in the other categories of manifolds and one has bijections
- , and, and
It is well known that the spaces
- , and
are mutually not homotopy equivalent and hence one obtains three different cohomology theories.
Sullivan analyzed the cases and . He showed that these spaces possess alternative inifinite loop space structures which are in fact better from the following point of view: Recall that there is a surgery obstruction map from normal invariants to the L-group. With the above described groups structure on the normal invariants this map is NOT a homomorphism. However, with the group structure from Sullivan's theorem it becomes a homomorphism in the categories, and . His theorem also links these new group structures to the well-known cohomology theories: the singular cohomology and real K-theory.
Read more about this topic: Normal Invariant
Other articles related to "categories":
... Landscape architecture categories - Landscape design history categories - Landscape gardens categories - Landscape or garden designers categories - Landscape categories - Lists of plants categories - List of ...
... generally divide fish and other sea creatures into three categories, analogous to conservation status categories Red ("say no" or "avoid") Yellow or orange ("think twice", "good ... The categories do not simply reflect the imperilment of individual species, but also consider the environmental impacts of how and where they are fished, such as through bycatch or ocean bottom trawlers ...
... the peoples of SARC countries, decided that certain categories of dignitaries should be entitled to a Special Travel document, which would exempt them from visas within the region ... kept under review the list of entitled categories ... Currently the list included 24 categories of entitled persons, which include Dignitaries, Judges of higher courts, Parliamentarians, Senior Officials, Businessmen, Journalists, Sportsmen etc ...
... In Brazil, the term "preto" (black) is one of the five categories used by the Brazilian Census, along with "branco" (White), "pardo" (Multiracial, brown), "amarelo" (yellow, East Asian ... of socioeconomic indicators, have been considering the categories "preto" and "pardo" together, as a single category called "negro" (Black, capital initial), since the indicators ... In the July 1998 PME, the categories "Afro-Brasileiro" (Afro-Brazilian) and "Africano Brasileiro" (African Brazilian) weren't used even once the category "Africano" (African) was used by 0.004 ...
Famous quotes containing the word categories:
“Kitsch ... is one of the major categories of the modern object. Knick-knacks, rustic odds-and-ends, souvenirs, lampshades, and African masks: the kitsch-object is collectively this whole plethora of “trashy,” sham or faked objects, this whole museum of junk which proliferates everywhere.... Kitsch is the equivalent to the “cliché” in discourse.”
—Jean Baudrillard (b. 1929)
“The analogy between the mind and a computer fails for many reasons. The brain is constructed by principles that assure diversity and degeneracy. Unlike a computer, it has no replicative memory. It is historical and value driven. It forms categories by internal criteria and by constraints acting at many scales, not by means of a syntactically constructed program. The world with which the brain interacts is not unequivocally made up of classical categories.”
—Gerald M. Edelman (b. 1928)