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| In the mathematical [[surgery theory]] the '''surgery exact sequence''' is the main technical tool to calculate the [[surgery structure set]] of a compact [[manifold]] in dimension <math>>4</math>. The [[surgery structure set]] <math>\mathcal{S} (X)</math> of a compact <math>n</math>-dimensional manifold <math>X</math> is a pointed set which classifies <math>n</math>-dimensional manifolds within the homotopy type of <math>X</math>.
| | It involves expertise and knowledge of various tools and technologies used for creating websites. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. Should you go with simple HTML or use a platform like Wordpress. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site. <br><br>The Internet is a vast open market where businesses and consumers congregate. Some of the Wordpress development services offered by us are:. It sorts the results of a search according to category, tags and comments. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. Akismet is really a sophisticated junk e-mail blocker and it's also very useful thinking about I recieve many junk e-mail comments day-to-day across my various web-sites. <br><br>ve labored so hard to publish and put up on their website. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. After age 35, 18% of pregnancies will end in miscarriage. Nonetheless, with stylish Facebook themes obtainable on the Globe Broad Internet, half of your enterprise is done previously. If you treasured this article and you would like to obtain more info concerning [http://dinky.in/?WordpressDropboxBackup408094 wordpress dropbox backup] kindly visit the web page. Websites using this content based strategy are always given top scores by Google. <br><br>If all else fails, please leave a comment on this post with the issue(s) you're having and help will be on the way. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. The templates are designed to be stand alone pages that have a different look and feel from the rest of your website. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white. <br><br>Many developers design websites and give them to the clients, but still the client faces problems to handle the website. s ability to use different themes and skins known as Word - Press Templates or Themes. The days of spending a lot of time and money to have a website built are long gone. Working with a Word - Press blog and the appropriate cost-free Word - Press theme, you can get a professional internet site up and published in no time at all. 95, and they also supply studio press discount code for their clients, coming from 10% off to 25% off upon all theme deals. |
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| The basic idea is that in order to calculate <math>\mathcal{S} (X)</math> it is enough to understand the other terms in the sequence, which are usually easier to determine. These are on one hand the [[normal invariants]] which form [[Cohomology#Cohomology theories|generalized cohomology groups]], and hence one can use standard tools of algebraic topology to calculate them at least in principle. On the other hand there are the [[L-theory|L-groups]] which are defined algebraically in terms of [[quadratic forms]] or in terms of chain complexes with quadratic structure. A great deal is known about these groups. Another part of the sequence are the [[surgery obstruction]] maps from normal invariants to the L-groups. For these maps there are certain [[characteristic classes]] formulas, which enable to calculate them in some cases. Knowledge of these three components, that means: the normal maps, the L-groups and the surgery obstruction maps is enough to determine the structure set (at least up to extension problems).
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| In practice one has to proceed case by case, for each manifold <math>\mathcal{} X</math> it is a unique task to determine the surgery exact sequence, see some examples below. Also note that there are versions of the surgery exact sequence depending on the category of manifolds we work with: smooth (DIFF), PL, or topological manifolds and whether we take Whitehead torsion into account or not (decorations <math>s</math> or <math>h</math>).
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| The original 1962 work of [[William Browder (mathematician)|Browder]] and [[Sergei Novikov (mathematician)|Novikov]] on the existence and uniqueness of manifolds within a [[simply-connected]] homotopy type was reformulated by [[Dennis Sullivan|Sullivan]] in 1966 as a '''surgery exact sequence'''.
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| In 1970 [[C.T.C. Wall|Wall]] developed [[non-simply-connected]] surgery theory and the surgery exact sequence for manifolds with arbitrary fundamental group.
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| ==Definition==
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| The surgery exact sequence is defined as | |
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| : <math>
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| \cdots \to \mathcal{N}_\partial (X \times I) \to L_{n+1} (\pi_1 (X)) \to \mathcal{S}(X) \to \mathcal{N} (X) \to L_n (\pi_1 (X))
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| </math>
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| where:
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| the entries <math>\mathcal{N}_\partial (X \times I)</math> and <math>\mathcal{N} (X)</math> are the abelian groups of [[normal invariants]],
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| the entries <math>\mathcal{} L_{n+1} (\pi_1 (X)) </math> and <math>\mathcal{} L_{n} (\pi_1 (X)) </math> are the [[L-theory|L-groups]] associated to the group ring <math>\mathbb{Z}[\pi_1 (X)]</math>,
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| the maps <math>\theta \colon \mathcal{N}_\partial (X \times I) \to L_{n+1} (\pi_1 (X))</math> and <math>\theta \colon \mathcal{N} (X) \to L_n (\pi_1 (X))</math> are the [[surgery obstruction]] maps,
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| the arrows <math>\partial \colon L_{n+1} (\pi_1 (X)) \to \mathcal{S}(X)</math> and <math>\eta \colon \mathcal{S}(X) \to \mathcal{N} (X)</math> will be explained below.
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| ==Versions==
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| There are various versions of the surgery exact sequence. One can work in either of the three categories of manifolds: differentiable (smooth), PL, topological. Another possibility is to work with the decorations <math>s</math> or <math>h</math>.
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| ==The entries==
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| ===Normal invariants===
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| {{main|Normal invariants}}
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| A degree one normal map <math>(f,b) \colon M \to X</math> consists of the following data: an <math>n</math>-dimensional oriented closed manifold <math>M</math>, a map <math>f</math> which is of degree one (that means <math>f_* ([M]) = [X]</math>, and a bundle map <math>b \colon TM \oplus \varepsilon^k \to \xi</math> from the stable tangent bundle of <math>M</math> to some bundle <math>\xi</math> over <math>X</math>. Two such maps are equivalent if there exists a normal bordism between them (that means a bordism of the sources covered by suitable bundle data). The equivalence classes of degree one normal maps are called '''normal invariants'''.
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| When defined like this the normal invariants <math>\mathcal{N} (X)</math> are just a pointed set, with the base point given by <math>(id,id)</math>. However the [[Pontrjagin-Thom construction#Framed cobordism|Pontrjagin-Thom]] construction gives <math>\mathcal{N} (X)</math> a structure of an abelian group. In fact we have a non-natural bijection
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| : <math>\mathcal{N} (X) \cong [X,G/O]</math>
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| where <math>G/O</math> denotes the homotopy fiber of the map <math>J \colon BO \to BG</math>, which is an infinite loop space and hence maps into it define a generalized cohomology theory. There are corresponding identifications of the normal invariants with <math>[X,G/PL]</math> when working with PL-manifolds and with <math>[X,G/TOP]</math> when working with topological manifolds.
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| ===L-groups===
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| {{main|L-theory}}
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| The <math>L</math>-groups are defined algebraically in terms of [[quadratic forms]] or in terms of chain complexes with quadratic structure. See the main article for more details. Here only the properties of the L-groups described below will be important.
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| ===Surgery obstruction maps===
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| {{main|Surgery obstruction}}
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| The map <math>\theta \colon \mathcal{N} (X) \to L_n (\pi_1 (X))</math> is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property (when <math>n \geq 5</math>:
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| A degree one normal map <math>(f,b) \colon M \to X</math> is normally cobordant to a homotopy equivalence if and only if the image <math>\theta (f,b)=0</math> in <math>L_n (\mathbb{Z} [\pi_1 (X)])</math>.
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| ===The normal invariants arrow <math>\eta \colon \mathcal{S}(X) \to \mathcal{N} (X)</math>===
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| Any homotopy equivalence <math>f \colon M \to X</math> defines a degree one normal map.
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| ===The surgery obstruction arrow <math>\partial \colon L_{n+1} (\pi_1 (X)) \to \mathcal{S}(X)</math>===
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| This arrow describes in fact an action of the group <math>L_{n+1} (\pi_1 (X))</math> on the set <math>\mathcal{S}(X)</math> rather than just a map. The definition is based on the realization theorem for the elements of the <math>L</math>-groups which reads as follows:
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| Let <math>M</math> be an <math>n</math>-dimensional manifold with <math>\pi_1 (M) \cong \pi_1 (X)</math> and let <math>x \in L_{n+1} (\pi_1 (X))</math>. Then there exists a degree one normal map of manifolds with boundary
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| : <math>(F,B) \colon (W,M,M') \to (M \times I, M \times 0, M \times 1)</math>
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| with the following properties: | |
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| 1. <math>\theta (F,B) = x \in L_{n+1} (\pi_1 (X))</math>
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| 2. <math>F_0 \colon M \to M \times 0</math> is a diffeomorphism
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| 3. <math>F_1 \colon M' \to M \times 1</math> is a homotopy equivalence of closed manifolds
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| Let <math>f \colon M \to X</math> represent an element in <math>\mathcal{S} (X)</math> and let <math>x \in L_{n+1} (\pi_1 (X))</math>. Then <math>\partial (f,x)</math> is defined as <math>f \circ F_1 \colon M' \to X</math>.
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| ==The exactness==
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| Recall that the surgery structure set is only a pointed set and that the surgery obstruction map <math>\theta</math> might not be a homomorphism. Hence it is necessary to explain what is meant when talking about the "exact sequence". So the surgery exact sequence is an exact sequence in the following sense:
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| For a normal invariant <math>z \in \mathcal{N} (X)</math> we have <math>z \in \mathrm{Im} (\eta)</math> if and only if <math>\theta (z) = 0</math>. For two manifold structures <math>x_1, x_2 \in \mathcal{S} (X)</math> we have <math>\eta (x_1) = \eta(x_2)</math> if and only if there exists <math>u \in L_{n+1} (\pi_1 (X))</math> such that <math>\partial (u,x_1) = x_2</math>. For an element <math>u \in L_{n+1} (\pi_1 (X))</math> we have <math>\partial (u,\mathrm{id}) = \mathrm{id}</math> if and only if <math>u \in \mathrm{Im} (\theta)</math>.
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| ==Versions revisited==
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| In the topological category the surgery obstruction map can be made into a homomorphism. This is achieved by putting an alternative abelian group structure on the normal invariants as described [[Normal invariants|here]]. Moreover, the surgery exact sequence can be identified with the algebraic surgery exact sequence of Ranicki which is an exact sequence of abelian groups by definition. This gives the structure set <math>\mathcal{S} (X)</math> the structure of an abelian group. Note, however, that there is to this date no satisfactory geometric description of this abelian group structure.
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| ==Classification of manifolds==
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| The answer to the organizing questions of the [[surgery theory]] can be formulated in terms of the surgery exact sequence. In both cases the answer is given in the form of a two-stage obstruction theory.
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| The existence question. Let <math>X</math> be a finite Poincaré complex. It is homotopy equivalent to a manifold if and only if the following two conditions are satisfied. Firstly, <math>X</math> must have a vector bundle reduction of its Spivak normal fibration. This condition can be also formulated as saying that the set of normal invariants <math>\mathcal{N} (X)</math> is non-empty. Secondly, there must be a normal invariant <math>x \in \mathcal{N} (X)</math> such that <math>\theta (x) = 0</math>. Equivalently, the surgery obstruction map <math>\theta \colon \mathcal{N} (X) \rightarrow L_{n} (\pi_1 (X))</math> hits <math>0 \in L_{n} (\pi_1 (X))</math>.
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| The uniqueness question. Let <math>f \colon M \to X</math> and <math>f' \colon M' \to X</math> represent two elements in the [[surgery structure set]] <math>\mathcal{S} (X)</math>. The question whether they represent the same element can be answered in two stages as follows. First there must be a normal cobordism between the degree one normal maps induced by <math>\mathcal{} f</math> and <math>\mathcal{} f'</math>, this means <math>\mathcal{} \eta (f) = \eta (f')</math> in <math>\mathcal{N} (X)</math>. Denote the normal cobordism <math>(F,B) \colon (W,M,M') \to (X \times I, X \times 0, X \times 1)</math>. If the surgery obstruction <math>\mathcal{} \theta (F,B)</math> in <math>\mathcal{} L_{n+1} (\pi_1 (X))</math> to make this normal cobordism to an [[h-cobordism]] (or [[s-cobordism]]) relative to the boundary vanishes then <math>\mathcal{} f</math> and <math>\mathcal{} f'</math> in fact represent the same element in the [[surgery structure set]].
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| ==Examples==
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| ===1. [[Homotopy sphere]]s===
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| This is an example in the smooth category, <math>n \geq 5</math>.
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| The idea of the surgery exact sequence is implicitly present already in the original article of Kervaire and Milnor on the groups of homotopy spheres. In the present terminology we have
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| : <math>\mathcal{S}^{DIFF} (S^n) = \Theta^n</math>
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| <math>\mathcal{N}^{DIFF} (S^n) = \Omega^{alm}_n</math> the cobordism group of almost framed <math>n</math> manifolds, <math>\mathcal{N}^{DIFF}_\partial (S^n \times I) = \Omega^{alm}_{n+1}</math>
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| <math>L_n (1) =\mathbb{Z}, 0, \mathbb{Z}_2, 0</math> where <math>n \equiv 0,1,2,3</math> mod <math>4</math> (recall the <math>4</math>-periodicity of the [[L-theory|L-groups]])
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| The surgery exact sequence in this case is an exact sequence of abelian groups. In addition to the above identifications we have
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| <math>bP^{n+1} = \mathrm{ker} (\eta \colon \mathcal{S}^{DIFF} (S^n) \to \mathcal{N}^{DIFF} (S^n)) = \mathrm{coker} (\theta \colon \mathcal{N}^{DIFF}_\partial (S^n \times I) \to L_{n+1} (1))</math>
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| Because the odd-dimensional L-groups are trivial one obtains these exact sequences:
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| : <math>0 \to \Theta^{4i} \to \Omega^{alm}_{4i} \to \mathbb{Z} \to bP^{4i} \to 0</math>
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| : <math>0 \to \Theta^{4i-2} \to \Omega^{alm}_{4i-2} \to \mathbb{Z}/2 \to bP^{4i-2} \to 0</math>
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| : <math>0 \to bP^{2j} \to \Theta^{2j-1} \to \Omega^{alm}_{2j-1} \to 0</math>
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| The results of Kervaire and Milnor are obtained by studying the middle map in the first two sequences and by relating the groups <math>\Omega_i^{alm}</math> to stable homotopy theory.
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| ===2. Topological spheres===
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| The [[generalized Poincaré conjecture]] in dimension <math>n</math> can be phrased as saying that <math>\mathcal{S}^{TOP} (S^n) = 0</math>. It has been proved for any <math>n</math> by the work of Smale, Freedman and Perelman. From the surgery exact sequence for <math>S^n</math> for <math>n \geq 5</math> in the topological category we see that | |
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| : <math>\theta \colon \mathcal{N}^{TOP} (S^n) \to L_n (1)</math>
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| is an isomorphism. (In fact this can be extended to <math>n \geq 1</math> by some ad-hoc methods.)
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| ===3. Complex [[projective space]]s in the topological category===
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| The complex projective space <math>\mathbb{C} P^n</math> is a <math>(2n)</math>-dimensional topological manifold with <math>\pi_1 (\mathbb{C} P^n)=1</math>. In addition it is known that in the case <math>\pi_1 (X) =1</math> in the topological category the surgery obstruction map <math>\theta</math> is always surjective. Hence we have
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| : <math>0 \to \mathcal{S}^{TOP} (\mathbb{C} P^n) \to \mathcal{N}^{TOP} (\mathbb{C} P^n) \to L_{2n}(1) \to 0</math>
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| From the work of Sullivan one can calculate
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| : <math>\mathcal{N} (\mathbb{C} P^n) \cong \oplus_{i=1}^{\lfloor n/2 \rfloor} \mathbb{Z} \oplus \oplus_{i=1}^{\lfloor (n+1)/2 \rfloor} \mathbb{Z}_2</math> and hence <math>\mathcal{S} (\mathbb{C} P^n) \cong \oplus_{i=1}^{\lfloor (n-1)/2 \rfloor} \mathbb{Z} \oplus \oplus_{i=1}^{\lfloor n/2 \rfloor} \mathbb{Z}_2</math>
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| ===4. [[Aspherical space|Aspherical]] manifolds in the topological category===
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| An aspherical <math>n</math>-dimensional manifold <math>X</math> is an <math>n</math>-manifold such that <math>\pi_i (X) = 0</math> for <math>i \geq 2</math>. Hence the only non-trivial homotopy group is <math>\pi_1 (X)</math>
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| One way to state the [[Borel conjecture]] is to say that for such <math>X</math> we have that the [[Whitehead group]] <math>Wh (\pi_1 (X))</math> is trivial and that
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| : <math>\mathcal{S} (X) = 0</math>
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| This conjecture was proven in many special cases - for example when <math>\pi_1 (X)</math> is <math>\mathbb{Z}^n</math>, when it is the fundamental group of a negatively curved manifold or when it is a word-hyperbolic group or a CAT(0)-group.
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| The statement is equivalent to showing that the surgery obstruction map to the right of the surgery structure set is injective and the surgery obstruction map to the left of the surgery structure set is surjective. Most of the proofs of the above mentioned results are done by studying these maps or by studying the [[assembly map]]s with which they can be identified. See more details in [[Borel conjecture]], [[Farrell-Jones Conjecture]]. | |
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| ==References==
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| *{{Citation | last1=Browder | first1=William | title=Surgery on simply-connected manifolds | publisher=[[Springer-Verlag]] | location=Berlin, New York | id={{MathSciNet | id = 0358813}} | year=1972}}
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| *{{Citation | last1=Lück | first1=Wolfgang | title=A basic introduction to surgery theory | url=http://131.220.77.52/lueck/data/ictp.pdf | publisher= ICTP Lecture Notes Series 9, Band 1, of the school "High-dimensional manifold theory" in Trieste, May/June 2001, Abdus Salam International Centre for Theoretical Physics, Trieste 1-224 | year=2002}}
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| *{{Citation | last1=Ranicki | first1=Andrew | title = Algebraic L-theory and topological manifolds | publisher=[[Cambridge University Press]] | series = Cambridge Tracts in Mathematics | url = http://www.maths.ed.ac.uk/~aar/books/topman.pdf | year = 1992| volume=102 }}
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| *{{Citation | last1=Ranicki | first1=Andrew | title=Algebraic and Geometric Surgery | url=http://www.maths.ed.ac.uk/~aar/books/surgery.pdf | publisher=Oxford Mathematical Monographs, Clarendon Press | isbn=978-0-19-850924-0 | id={{MathSciNet | id = 2061749}} | year=2002}}
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| *{{Citation | last1=Wall | first1=C. T. C. | title=Surgery on compact manifolds | publisher=[[American Mathematical Society]] | location=Providence, R.I. | edition=2nd | series=Mathematical Surveys and Monographs | isbn=978-0-8218-0942-6 | id={{MathSciNet | id = 1687388}} | year=1999 | volume=69}}
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| [[Category:Surgery theory]]
| |
It involves expertise and knowledge of various tools and technologies used for creating websites. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. Should you go with simple HTML or use a platform like Wordpress. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site.
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