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| | Hi, everybody! My name is Tracee. <br>It is a little about myself: I live in Italy, my city of Spigno Monferrato. <br>It's called often Eastern or cultural capital of AL. I've married 4 years ago.<br>I have two children - a son (Rachelle) and the daughter (Emely). We all like Computer programming.<br><br>Also visit my webpage: [http://www.examiner.com/article/toothache-remedies-home-remedy-for-toothache Home Remedy for Toothache] |
| In [[mathematics]], '''assembly maps''' are an important concept in [[geometric topology]]. From the [[homotopy]]-theoretical viewpoint, an assembly map is a [[Universal property|universal]] approximation of a homotopy invariant [[functor]] by a [[homology theory]] from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data.
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| Assembly maps for [[algebraic K-theory]] and [[L-theory]] play a central role in the topology of high-dimensional [[manifold]]s, since their [[homotopy fiber]]s have a direct geometric interpretation. [[Equivariant map|Equivariant]] assembly maps are used to formulate the [[Farrell–Jones conjecture]]s in K- and L-theory.
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| ==Homotopy-theoretical viewpoint==
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| It is a classical result that for any generalized [[homology theory]] <math>h_*</math> on the [[category of topological spaces]] (assumed to be homotopy equivalent to [[CW-complex]]es), there is a [[spectrum (homotopy theory)|spectrum]] <math>E</math> such that
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| :<math>h_*(X)\cong \pi_*(X_+\wedge E),</math> | |
| where <math>X_+:=X\coprod \{*\}</math>.
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| The functor <math>X\mapsto X_+ \wedge E</math> from spaces to spectra has the following properties:
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| * It is homotopy-invariant (preserves homotopy equivalences). This reflects the fact that <math>h_*</math> is homotopy-invariant.
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| * It preserves homotopy co-cartesian squares. This reflects that fact that <math>h_*</math> has [[Mayer-Vietoris sequence]]s, an equivalent characterization of excision.
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| * It preserves arbitrary [[coproduct]]s. This reflects the disjoint-union axiom of <math>h_*</math>.
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| A functor from spaces to spectra fulfilling these properties is called '''excisive'''.
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| Now suppose that <math>F</math> is a homotopy-invariant, not necessarily excisive functor. An assembly map is a [[natural transformation]] <math>\alpha\colon F^\%\to F</math> from some excisive functor <math>F^\%</math> to <math>F</math> such that <math>F^\%(*)\to F(*)</math> is a homotopy equivalence.
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| If we denote by <math>h_*:=\pi_*\circ F^\%</math> the associated homology theory, it follows that the induced natural transformation of graded [[abelian group]]s <math>h_*\to \pi_*\circ F</math> is the universal transformation from a homology theory to <math>\pi_*\circ F</math>, i.e. any other transformation <math>k_*\to\pi_*\circ F</math> from some homology theory <math>k_*</math> factors uniquely through a transformation of homology theories <math>k_*\to h_*</math>.
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| Assembly maps exist for any homotopy invariant functor, by a simple homotopy-theoretical construction.
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| ==Geometric viewpoint==
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| As a consequence of the [[Mayer-Vietoris sequence]], the value of an excisive functor on a space <math>X</math> only depends on its value on 'small' subspaces of <math>X</math>, together with the knowledge how these small subspaces intersect. In a cycle representation of the associated homology theory, this means that all cycles must be representable by small cycles. For instance, for [[singular homology]], the excision property is proved by subdivision of [[simplex|simplices]], obtaining sums of small simplices representing arbitrary homology classes.
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| In this spirit, for certain homotopy-invariant functors which are not excisive, the corresponding excisive theory may be constructed by imposing 'control conditions', leading to the field of [[controlled topology]]. In this picture, assembly maps are 'forget-control' maps, i.e. they are induced by forgetting the control conditions.
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| ==Importance in geometric topology==
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| Assembly maps are studied in geometric topology mainly for the two functors <math>L(X)</math>, algebraic [[L-theory]] of <math>X</math>, and <math>A(X)</math>, [[algebraic K-theory]] of spaces of <math>X</math>. In fact, the homotopy fibers of both assembly maps have a direct geometric interpretation when <math>X</math> is a compact topological manifold. Therefore knowledge about the geometry of compact topological manifolds may be obtained by studying <math>K</math>- and <math>L</math>-theory and their respective assembly maps.
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| In the case of <math>L</math>-theory, the homotopy fiber <math>L_\%(M)</math> of the corresponding assembly map <math> L^\%(M)\to L(M)</math>, evaluated at a compact topological manifold <math>M</math>, is homotopy equivalent to the space of block structures of <math>M</math>. Moreover, the fibration sequence
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| :<math> L_\%(M)\to L^\%(M)\to L(M)</math> | |
| induces a [[long exact sequence]] of homotopy groups which may be identified with the [[surgery exact sequence]] of <math>M</math>. This may be called the '''fundamental theorem of surgery theory''' and was developed subsequently by Browder, Novikov, Sullivan, Wall, Quinn, and Ranicki.
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| For <math>A</math>-theory, the homotopy fiber <math>A_\%(M)</math> of the corresponding assembly map is homotopy equivalent to the space of stable [[h-cobordism]]s on <math>M</math>. This fact is called the '''stable parametrized h-cobordism theorem''', proven by Waldhausen-Jahren-Rognes. It may be viewed as a parametrized version of the classical theorem which states that equivalence classes of h-cobordisms on <math>M</math> are in 1-to-1 correspondence with elements in the [[Whitehead group]] of <math>\pi_1(M)</math>.
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| [[Category:Surgery theory]]
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| [[Category:K-theory]]
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Hi, everybody! My name is Tracee.
It is a little about myself: I live in Italy, my city of Spigno Monferrato.
It's called often Eastern or cultural capital of AL. I've married 4 years ago.
I have two children - a son (Rachelle) and the daughter (Emely). We all like Computer programming.
Also visit my webpage: Home Remedy for Toothache