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Topic: Homotopy invariant

	Homotopy - Wikipedia, the free encyclopedia
		An outstanding use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.
		An example of an algebraic invariant of topological spaces which is not homotopy-invariant is compactly supported homology (which is, roughly speaking, the homology of the compactification, and compactification is not homotopy-invariant).
		An example of a homotopy invariant is the fundamental group of a space, already mentioned earlier.
	en.wikipedia.org /wiki/Homotopy (1019 words)

	[No title]
		But after replacing E up to homotopy by a space X such that the new map p : X i B is a fibration, the space p-1(), called the homotopy* fiber of f, becomes a unique homotopy invariant of f.
		B of homotopy fiber F, your best chance for computing the algebra H*(F) is to apply the formidable machinery of the Eilenberg-Moore or Serre spectral sequences using all their algebraic structure.
		_ A(B) C In particular, the cohomology algebra of the homotopy fiber of f, H(F), is isomorphic to the cohomology of the homotopy* cofiber of A(f), H*(_A(B) C).
	hopf.math.purdue.edu /Menichi/Cohomology_Fiber.txt (11127 words)

	Homotopy biography .ms (Site not responding. Last check: 2007-11-05)
		In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
		Given two spaces X and Y, we say they are homotopy equivalent or of the same homotopy type if there exist continuous maps f : X → Y and g : Y → X such that g o f is homotopic to the identity map id
		A typical homotopy invariant is the fundamental group of a space, already mentioned earlier.
	homotopy.biography.ms (734 words)

	[No title]
		The purpose of this paper is to show that a suitable iterated suspension of F (M, k) is a homotopy invariant.
		N is a homotopy equivalence such that composition K !h N M is homotopic to f.
		Proof of Theorem A Suppose that M and N are homotopy equivalent r-connected (r 0) closed PL manifolds of dimension d.
	hopf.math.purdue.edu /Aouina-Klein/config_stable.txt (2455 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		We also show that the homotopy categories associated to the two categories are equivalent to the homotopy categories of simplicial presheaves and homotopy 2-types, respectively.
		This invariant can be easily calculated from the current picture of wave fronts on M without the knowledge of the propagation law for the wave fronts.
		David J. Pengelley (davidp@nmsu.edu) Frank Williams (frank@nmsu.edu) The algebra S of symmetric invariants over the field with two elements is an unstable algebra over the Steenrod algebra A and is isomorphic to the mod two cohomology of BO, the classifying space for vector bundles.
	claude.math.wesleyan.edu /~mhovey/archive/all03 (11093 words)

	Citebase - Homotopy Algebras are Homotopy Algebras (Site not responding. Last check: 2007-11-05)
		We prove that strongly homotopy algebras (such as $A_\infty$, $C_\infty$, sh Lie, $B_\infty$, $G_\infty$,...) are homotopically invariant in the category of chain complexes.
		We present a definition of homotopy algebra for an operad, and explore its consequences.
		It is shown that any compact K\"ahler manifold $M$ gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the Dolbeault complex.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:math/9907138 (864 words)

	Motivic Homotopy Theory Program (Site not responding. Last check: 2007-11-05)
		In the second we will study properties of homotopy invariant cross functors on the category of schemes including a general duality theorem.
		The motivic homotopy theory is the homotopy theory for algebraic varieties and, more generally, for Grothendieck's schemes which is based on the analogy between the affine line and the unit interval.
		Eventually, the motivic homotopy theory is expected to provide techniques which may help to solve problems in algebraic geomerty such as various "standard conjectures on algebraic cycles", Beilinson-Soule vanishing and rigidity conjectures, the Bloch-Kato conjecture etc.
	www.math.ias.edu /~vladimir/seminar.html (870 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		In their book "Homotopy invariant structures on topological structures", Boardman and Vogt construct from any topological operad O a new one WO, which can be thought of as a "weakened" version of O in which all the laws of O now hold only up to homotopy in a coherent way.
		For example, Boardman and Vogt point out that their construction can be used to discuss homotopy colimits, which makes me wonder if the construction of WO from O could be done slickly using homotopy colimits.
		> > In their book "Homotopy invariant structures on topological structures", > Boardman and Vogt construct from any topological operad O a new one WO, > which can be thought of as a "weakened" version of O in which all the > laws of O now hold only up to homotopy in a coherent way.
	www.mta.ca /~cat-dist/catlist/1999/weak-op (307 words)

	MFO
		We define a refinement of the homotopy relation which is generated by composing maps with homotopy trivial self equivalences of the objects.
		It holds that a surjective homomorphism of simplicial loops is a Kan fibration, the Moore complex is a loop-valued chain complex with homology the homotopy groups of the simplicial loop, and the simplicial loop is minimal iff the Moore complex is minimal.
		As an application, we show that inversion invariant cohomology is isomorphic to the usual group cohomology if the coefficients are local away from 2.
	www.mfo.de /organisation/institute/klaus/skhomepage.html (2346 words)

	ScienceDaily Third Party New : Homotopy Invariant Algebraic Structures: A Conference in Honor of Mike Boardman : Ams ... (Site not responding. Last check: 2007-11-05)
		Third Party New : Homotopy Invariant Algebraic Structures: A Conference in Honor of Mike Boardman : Ams Special Session on Homotopy Theory, January 1998, Baltimore, MD (Contemporary Mathematics)
		Homotopy Invariant Algebraic Structures: A Conference in Honor of Mike Boardman : Ams Special Session on Homotopy Theory, January 1998, Baltimore, MD (Contemporary Mathematics)
		Powerful Mineral Mapper Heads To Mars (August 14, 2005) — With the launch of NASA's Mars Reconnaissance Orbiter spacecraft, the Compact Reconnaissance Imaging Spectrometer for Mars -- or CRISM -- joins the set of high-tech detectives seeking traces of water on the red planet.
	www.sciencedaily.com /cgi-bin/apf4/amazon_products_feed.cgi?myOperation=New&ItemId=082181057X (1440 words)

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