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Topic: Homology theories

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Extraordinary cohomology theory

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	PlanetMath: homology
		Homology is the general name for a number of functors from topological spaces to abelian groups (or more generally modules over a fixed ring).
		It is based on computing the homology groups of a simplicial complex (generally a finite one).
		This is version 14 of homology, born on 2002-12-10, modified 2007-03-29.
	planetmath.org /encyclopedia/HomologyTopologicalSpace.html (519 words)

	- The True.Origin Archive -
		Evolutionary Theories on Gender and Sexual Reproduction (Harrub and Thompson)—manifests the shortcomings of evolutionary theories on the origin of sex.
		Homology in Biology—A Problem for Naturalistic Science (Jonathan Wells)—shows how without an empirically demonstrated naturalistic mechanism to account for homology, design remains a possibility deniable only on the basis of questionable philosophical assumptions.
		The Tablet Theory of Genesis Authorship (Curt Sewell)—presents the “tablet theory” concerning the origin of the written record of Genesis, affirming biblical reliability and man’s inherent literary nature.
	www.trueorigin.org (2711 words)

	[No title]
		A homology theory on a triangulated category S is an exact functor to an Abelian category which preser* ves the coproducts that exist in S. Unless we state otherwise, the target category *will always be taken to be the category Ab of Abelian groups.
		A cohomology theory with values in an Abelian category B is a homology theory with values in Bop.
		The subcategory consisting of towers of finite spectra is equivalent to the opposit* e of the category of homology* theories with countable coefficients.
	hopf.math.purdue.edu /Christensen-Strickland/phantoms.txt (10943 words)

	Journal of Dinosaur Paleontology
		Original commentary on the recent spate of bills mandating evolution not be taught as fact, focusing on the definitions of fact, theory, etc., and the need for and motivation behind such legislation.
		Discussion of the age of the largest impact crater in the continental U.S., and its possible affect on impact extinction theory.
		Article detailing a new theory of why the impact of an asteroid at Chicxulub in the Yucatan 65 million years ago seems to have more adversely affected the animals of North America than elsewhere.
	www.dinosauria.com /jdp/jdp.htm (1618 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		Our point in making this rigorous is that both the category of homology theories and the bimodule of phantom maps are determined by the category of finite spectra, and so we see that the category of spectra is determined up to extension by the category of finite spectra.
		A homology theory on a triangulated category S is an exact functor to an Abelian category which preserves the coproducts that exist in S. Unless we state otherwise, the target category will always be taken to be the category Ab of Abelian groups.
		It is shown in [15, Section 4] that a homology theory defined on F has an essentially unique extension to a homology theory defined on all of S, so the categories of homology theories on F and S are equivalent.
	jdc.math.uwo.ca /papers/phantoms.txt (11288 words)

	Page Personnelle de Philippe Gaucher (Site not responding. Last check: 2007-10-16)
		Mathematically, this means that we investigate the combinatorics of the negative corner (or branching) homology of a globular $\omega$-category and the combinatorics of a new homology theory called the reduced branching homology.
		The latter is the homology of the quotient of the branching complex by the sub-complex generated by its thin elements.
		Some of the properties discovered so far on the corresponding simplicial homology theories are explained, in particular their links with geometric problems coming from concurrency theory in computer science.
	www.pps.jussieu.fr /~gaucher/enligne3.html (880 words)

	55: Algebraic topology
		Cohomology theories are a slight change from homology theories in that the directions of some homomorphisms are reversed; they're roughly the dual groups of the homology groups.
		Homology groups are particularly well suited to computation via some inductive procedure: if a space is somehow pieced together from simpler spaces (as unions, say, or fibrations) then the homology theories of the large space reflect those of the smaller spaces, together with some algebraic information which indicates the nature of the piecing-together.
		Apart from homology groups and their kin, the principal algebraic tool used in topology is the set of homotopy groups of a space, and related concepts; in particular this includes the fundamental group (pi_1(X)) of a space.
	www.math.niu.edu /~rusin/known-math/index/55-XX.html (2581 words)

	Cornell Math - Thesis Abstracts (Topology) (Site not responding. Last check: 2007-10-16)
		We prove that the stabilizer of a vertex in the nerve is the mapping class group of a surface, and we identify stabilizers of higher-dimensional simplices with stabilizers of sets of conjugacy classes in $\fn$.
		By using Harer's homology stability theorems for mapping class groups to analyze the $d^1$ map, we find a bound on the dimension of the subspace of $H_(\out;\Q)$ generated by the stable rational mapping class homology*.
		In Chapter 5 generalized homology theories are shown to be very similar to standard, singular homology from the point of view of representations of mapping class groups.
	www.math.cornell.edu /~www/Research/Abstracts/topology.html (3195 words)

	[No title]
		Several new theories are introduced, including localizations for simplicial presheaves and presheaves of spectra at homology theories represented by presheaves of spectra, a theory of localization along a geometric topos morphism, and in particular a method of localizing a space or spectrum at a generalized \'etale cohomology theory.
		A standard procedure in the theory of C* algebras is to tensor a given algebra with the algebra of compact operators on a separable Hilbert space: this process is called ``stabilization'' (and is unrelated to homotopy-theoretic stablization).
		We show that the localization of $BG$ with respect to a multiplicative complex oriented homology theory $h_$ is again a space of type $K(\pi,1)$; in fact, it is the same as the localization of $BG$ with respect to the ordinary homology* theory determined by the ring $h_0$.
	claude.math.wesleyan.edu /~mhovey/archive/all97 (9156 words)

	AGT 1 (2001) Paper 13 (Abstract)
		In this paper we develop an axiomatic approach to coarse homology theories.
		We prove a uniqueness result concerning coarse homology theories on the category of `coarse CW-complexes'.
		Coarse geometry, exotic homology, coarse Baum-Connes conjecture, Novikov conjecture
	www.maths.warwick.ac.uk /agt/AGTVol1/agt-1-13.abs.html (81 words)

	K-theory Preprint Archives
		524: November 8, 2001, The obstruction to excision in K-theory and in cyclic homology, by Guillermo Cortiñas.
		329: February 3, 1999, On the descent proplem for topological cyclic homology and algebraic K-theory, by Stavros Tsalidis.
		324: December 18, 1998, Cyclic cohomology of Hopf algebras, and a non-commutative Chern-Weil theory, by Crainic Marius.
	www.mathematik.uni-osnabrueck.de /K-theory (11483 words)

	NORMAN STEENROD
		Together with Samuel Eilenberg, he axiomatized homology theory and established the modern framework of algebraic topology, a subject invented by Poincaré.
		The other difficulty was that homology was originally only understood for very restricted types of spaces (such as simplicial complexes); a theory that would work for general spaces was a natural objective.
		By the early 1940's, many homology theories had been defined, including those of Alexander and Veblen, Lefschetz, Vietoris, and Cech; it was unclear how these theories related to each other, and the subject was in a chaotic state.
	www.usna.edu /Users/math/meh/steenrod.html (704 words)

	Science - All About GOD
		- A theory in crisis in light of the tremendous advances we've made in molecular biology, biochemistry, genetics and information theory.
		What are the different theories on the origin of life?
		What is the theory of the extraterrestrial origin of life?
	www.allaboutgod.com /science.htm (1324 words)

	PeV Home Page (Site not responding. Last check: 2007-10-16)
		Homology of linear groups, Algebraic K-theory and connection with the Hilbert third Problem.
		Homology of the special linear group and K-theory, C. Acad.
		Homology of linear groups with coefficients in the adjoint action and K-theory, Preprint n
	www.math.univ-montp2.fr /~pev/index-eng.html (209 words)

	Vortragsankündigung (Site not responding. Last check: 2007-10-16)
		Zusammenfassung: In (algebraic) topology, homology and cohomology are basic tools to study the object of study.
		Besides ordinary homology, many generalized (co)homology theories are of important use.
		Special cases of this enrich theory classify line bundles with connection or gerbes, certain geometrical objects which are higher companions of vector bundles.
	www.mathematik.uni-muenchen.de /~mathkoll/vortraege/schick.html (132 words)

	Manchester Geometry Seminar (Site not responding. Last check: 2007-10-16)
		For many homology theories E(X) defined on topological spaces X, E(X) is more than an abelian group or even a module over E(point); it is also a comodule over the Hopf algebroid E(E).
		We show that for many of the common homology theories E, the category of comodules over E(E) depends up to equivalence only on the heights of E at the different primes.
		In particular, we give several important examples of homology theories E and F such that the categories of comodules over E(E) and F(F) are equivalent, even though the categories of modules over E(point) and F(point) are very different.
	www.ma.umist.ac.uk /tv/Seminar/2002-2003/hovey.html (136 words)

	Equipe de Géométrie Non commutative (Site not responding. Last check: 2007-10-16)
		Adapted homology theories have been defined and Fredholm properties obtained starting from a good definition of ellipticity (Bruening, Lesch, Shrohe, Schultz and others).
		Entire cyclic homology was defined by Connes as the natural target of a Chern charactrer of a bivariant $\theta$-summable Fredholm modules.
		First obtain a satisfactory and adaptable presentation of the various bivariant cyclic theories which seem appropriate targets for a Chern character, Secondly find a reasonnable bivariant version of Connes monovariant definiton of a $\theta $-summable Fredholm module and its Chern charactern, Finally establisch the multiplicative property of this construction.
	picard.ups-tlse.fr /~ncg/rech.php?lang=en (817 words)

	Pacific Journal of Mathematics - Abstract for 186-2-8 - Andreas Zastrow (Site not responding. Last check: 2007-10-16)
		The paper investigates a homology theory based on the ideas of Milnor and Thurston that by considering measures on the set of all singular simplices one should get alternate possibilities for describing the cycles of classical homology theory.
		It suggests slight changes to Milnor's and Thurston's original definitions (giving differences for wild topological spaces only) which ensure that their homology theory is well-defined on all topological spaces.
		An example showing that the coincidence between these both homology theories does not hold for all topological spaces is also included.
	nyjm.albany.edu:8000 /PacJ/1998/186-2-8.html (134 words)

	Birgit Richter; publications (Site not responding. Last check: 2007-10-16)
		Topological Hochschild homology of a ring $R$ is Hochschild homology of the cubical construction $Q_*(R)$.
		Gamma-homology of group algebras and of polynomial algebras, joint with Alan Robinson, in: Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic $K$-Theory, eds.: Paul Goerss and Stewart Priddy, Northwestern University, Editors, Cont.
		We consider the issue of passage from algebraic Galois extensions to topological ones applying obstruction theories of Robinson and Goerss-Hopkins to obtain topological models for algebraic Galois extensions and the necessary morphisms of S-algebras.
	www.math.uni-bonn.de /people/richter/publications.html (945 words)

	abstract 2002-02 (Site not responding. Last check: 2007-10-16)
		On the homology of algebras of Whitney functions.
		In this article we study several homology theories of the algebra
		It is shown that the periodic cyclic homology coincides with the de Rham cohomology, thus generalizing a result of Feigin-Tsygan.
	iml.univ-mrs.fr /editions/preprint2002/abstracts/abstract2002-02.html (157 words)

	[No title]
		To be precise, recall that Tate homology is a functor from $G$-spectra to $G$-spectra.
		We also develop a homotopical theory of $R$-ring spectra in $\sD_R$, analogous to the classical theory of ring spectra in the stable homotopy category, and we use it to give new constructions as $MU$-ring spectra of a host of fundamentally important spectra whose earlier constructions were both more difficult and less precise.
		Suppose that H is a multiplicative homology theory such that (1) both H (F P 1) and H (F P21) are free H (pt)-modules;and (2) the inclusion of the bottom cell Sd !F P 1 induces a monomorphism in the homology.
	claude.math.wesleyan.edu /~mhovey/archive/all95 (10957 words)

	UM Mathematics: (Site not responding. Last check: 2007-10-16)
		A very intriguing example of a homology theory is stable homotopy theory, which is most closely related to homotopy groups.
		The main point of these books is that operads can be used in constructing generalized cohomology theories outside the usual context of topological spaces.
		For example, in algebraic geometry, one can hope to understand 'integral mixed motives', which is conjectural notion of generalized (co)homology theories on algebraic varieties.
	www.math.lsa.umich.edu /people/faculty/krizI.shtml (365 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		Andrew Baker and Alain Jeanneret, Brave new Hopf algebroids and extensions of MU-algebras, Homology, Homotopy and Applications, vol.
		Andrei Lazarev, Homotopy theory of A_infty ring spectra and applications to MU-modules, K-Theory 24 (3): 243-281, November 2001.
		Whitehead, Generalized homology theories, Transactions of the American Mathematical Society, Vol.
	www.math.uio.no /~rognes/kurs/ma422h03 (289 words)

	Feature Article: Comparative Development of the Mammalian Isocortex and the Reptilian Dorsal Ventricular Ridge. ...
		Theories of Homology between the Dorsal Ventricular Ridge and the Isocortex
		Aboitiz F (1995) Homology in the evolution of the cerebral hemispheres: the case of reptilian dorsal ventricular ridge and its possible correspondence with mammalian neocortex.
		Striedter GF (1998) Progress in the study of brain evolution: from speculative theories to testable hypotheses.
	cercor.oxfordjournals.org /cgi/content/full/9/8/783 (6547 words)

	Dwyer: Strong convergence... (Site not responding. Last check: 2007-10-16)
		This paper proves that the ordinary homology Eilenberg-Moore spectral sequence (EMSS) of a fibration converges strongly if and only if the monodromy action of the fundamental group of the base on the homology of the fibre is nilpotent.
		The proof of the other implication proceeds by showing that the abutment of the EMSS fits into a spectral sequence equation: there is a Serre spectral sequence converging from the homology of the base with coefficients in the EMSS abutment to the homology of the total space.
		A technical point which comes up in the paper is that the EMSS abutment is most conveniently treated as a collection of towers of abelian groups, not as the collection of their inverse limits.
	www.nd.edu /~wgd/Html/Strong.Convergence.EMSS.html (382 words)

	L. Avramov Abstract (Site not responding. Last check: 2007-10-16)
		In this context, commutative algebras model morphisms of the underlying varieties, and homology theories of commutative algebras measure the singularities of the fibers of such morphisms.
		At the beginning of the talk it will be explained how classical results link important properties of commutative algebras, such as smoothness, to the vanishing of homology groups in the theories constructed by Gerhard Hochschild in the mid-1940's, and by Michel Andr\'e and Daniel Quillen in the late 1960's.
		This will be followed by a discussion of new homological characterizations of several classes of algebras in terms of vanishing of homology groups or finiteness of homology algebras.
	www.mth.msu.edu /~mccarthy/colloq.01-2/avramov-abstract.html (155 words)

	Prof. Matthias Kreck - Ruprecht-Karls-Universität Heidelberg (Site not responding. Last check: 2007-10-16)
		The most important tools for classification are homology theories.
		Apart from using for this the established homology theories, I have constructed various new homology theories.
		Recently, I was particularly attracted by a problem which is around for several decades, namely the question whether manifolds admitting a symmetry are less typical than those admitting no symmetry.
	www.mathi.uni-heidelberg.de /~kreck/Forschung.html (265 words)

	Voevodsky, V., Suslin, A., Friedlander, E.M.: Cycles, Transfers, and Motivic Homology Theories. (AM-143).
		The original goal that ultimately led to this volume was the construction of "motivic cohomology theory," whose existence was conjectured by A. Beilinson and S. Lichtenbaum.
		The theory of sheaves of relative cycles is developed in the first paper of this volume.
		The theory of presheaves with transfers and more specifically homotopy invariant presheaves with transfers is the main theme of the second paper.
	pup.princeton.edu /titles/7003.html (305 words)

	Paul Mitchener's Homepage
		In many cases a C*-algebra associated to a given geometric structure is defined by arbitrarily choosing a Hilbert space satisfying certain criteria and considering operators on that Hilbert space possessing properties determined by the geometry.
		This article is an exposition of the theory of simplicial sets and spaces, classifying spaces, homotopy colimits, the plus construction and the group completion theorem.
		This article is a very short summary of some aspects of the theory of symmetric spectra that I find useful in my work.
	www.uni-math.gwdg.de /mitch/research.html (1011 words)

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