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| In [[mathematics]], specifically in [[surgery theory]], the '''surgery obstructions''' define a map <math>\theta \colon \mathcal{N} (X) \to L_n (\pi_1 (X))</math> from the [[normal invariant]]s to the [[L-theory|L-group]]s which 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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| ==Sketch of the definition==
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| The surgery obstruction of a degree-one normal map has a relatively complicated definition.
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| Consider a degree-one normal map <math>(f,b) \colon M \to X</math>. The idea in deciding the question whether it is normally cobordant to a homotopy equivalence is to try to systematically improve <math>(f,b)</math> so that the map <math>f</math> becomes <math>m</math>-connected (that means the homotopy groups <math>\pi_* (f)=0</math> for <math>* \leq m</math>) for high <math>m</math>. It is a consequence of [[Poincaré duality]] that if we can achieve this for <math>m > \lfloor n/2 \rfloor</math> then the map <math>f</math> already is a homotopy equivalence. The word ''systematically'' above refers to the fact that one tries to do surgeries on <math>M</math> to kill elements of <math>\pi_i (f)</math>. In fact it is more convenient to use [[homology (mathematics)|homology]] of the [[universal cover]]s to observe how connected the map <math>f</math> is. More precisely, one works with the '''surgery kernels''' <math>K_i (\tilde M) : = \mathrm{ker} \{f_* \colon H_i (\tilde M) \rightarrow H_i (\tilde X)\}</math> which one views as <math>\mathbb{Z}[\pi_1 (X)]</math>-modules. If all these vanish, then the map <math>f</math> is a homotopy equivalence. As a consequence of Poincaré duality on <math>M</math> and <math>X</math> there is a <math>\mathbb{Z}[\pi_1 (X)]</math>-modules Poincaré duality <math>K^{n-i} (\tilde M) \cong K_i (\tilde M)</math>, so one only has to watch half of them, that means those for which <math>i \leq \lfloor n/2 \rfloor</math>.
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| Any degree-one normal map can be made <math>\lfloor n/2 \rfloor</math>-connected by the process called surgery below the middle dimension. This is the process of killing elements of <math>K_i (\tilde M)</math> for <math>i < \lfloor n/2 \rfloor</math> described [[Surgery theory|here]] when we have <math>p+q = n</math> such that <math>i = p < \lfloor n/2 \rfloor</math>. After this is done there are two cases.
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| 1. If <math>n=2k</math> then the only nontrivial homology group is the kernel <math>K_k (\tilde M) : = \mathrm{ker} \{f_* \colon H_k (\tilde M) \rightarrow H_k (\tilde X)\}</math>. It turns out that the cup-product pairings on <math>M</math> and <math>X</math> induce a cup-product pairing on <math>K_k(\tilde M)</math>. This defines a symmetric bilinear form in case <math>k=2l</math> and a skew-symmetric bilinear form in case <math>k=2l+1</math>. It turns out that these forms can be refined to <math>\varepsilon</math>-quadratic forms, where <math>\varepsilon = (-1)^k</math>. These <math>\varepsilon</math>-quadratic forms define elements in the L-groups <math>L_n (\pi_1 (X))</math>.
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| 2. If <math>n=2k+1</math> the definition is more complicated. Instead of a quadratic form one obtains from the geometry a quadratic formation, which is a kind of automorphism of quadratic forms. Such a thing defines an element in the odd-dimensional L-group <math>L_n (\pi_1 (X))</math>.
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| If the element <math>\theta (f,b)</math> is zero in the L-group surgery can be done on <math>M</math> to modify <math>f</math> to a homotopy equivalence.
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| Geometrically the reason why this is not always possible is that performing surgery in the middle dimension to kill an element in <math>K_k (\tilde M)</math> possibly creates an element in <math>K_{k-1} (\tilde M)</math> when <math>n = 2k</math> or in <math>K_{k} (\tilde M)</math> when <math>n=2k+1</math>. So this possibly destroys what has already been achieved. However, if <math>\theta (f,b)</math> is zero, surgeries can be arranged in such a way that this does not happen.
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| ==Example==
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| In the simply connected case the following happens.
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| If <math>n = 2k+1</math> there is no obstruction.
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| If <math>n = 4l</math> then the surgery obstruction can be calculated as the difference of the signatures of M and X.
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| If <math>n = 4l+2</math> then the surgery obstruction is the Arf-invariant of the associated kernel quadratic form over <math>\mathbb{Z}_2</math>.
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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://wwwmath.uni-muenster.de/reine/inst/lueck/homepages/wolfgang_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 and Geometric Surgery | url=http://www.maths.ed.ac.uk/~aar/books/index.htm | 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]]
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What a remarkable way to get rid of my experience at this school. Despite all the negativity I've found I had been able to work with a great teacher paid traffic . day. These types was the same one Got worked along with a few weeks ago on a Friday who had had a substitute pro. So again Identified it hard to figure out whether hints the children who were naturally obedient or whether the teacher had some .
Pumpkin Bean bag Toss: Two teams get their members take turns throwing two bean bags into a plastic pumpkin. Each team member end up being throw until he/she gets it in, then shows the bean bags to the other team member in line, and the like. The first team to have each player score is declared the winner.
A break clause allows the rights to revert to the writer after a certain period - usually few years after the sunday paper has been remaindered or goes out-of-print. It implies that a publisher does not retain the rights consistently.
Fill the feeder roughly about 1 and 1/2 inches with seed. However buy seed in bulk for about 19c per pound for sale. This will attract chickadees, cardinals, house finch, dark-eyed juncos, sparrows, starlings, blue jays and much of hungry squirrels.
GIFT Put. Create an image,keep in mind the length and width of gift its for, repeat the graphics. How many people can you think of who like their own gift wrap? Private householders maybe, to opt for the calendars, cards you are offering. But what of small shops, flower shops? Experience paper, legal paper, or roll paper for the larger wraps, in case the printer accepts it. Enjoyable.
Nightmare for your Bayou, located directly across the street from Houston's oldest cemetery, states to be "the only Haunted House in Houston can be really haunted!" Tickets are $30 different discounts and VIP passes available. Tend to be open from Friday, September 25 through Halloween furniture covers - http://www.36barstools.com/2011/10/halloween-furniture-covers,, starting at 7:30PM. They don't accept credit cards, but do a great onsite Atm machine. Bare feet, sandals or high heels are not suggested. Children under 10 must be accompanied by an personal. Pregnant women and those with asthma or heart conditions are not encouraged to venture to.
Don't embarrass your guests by preparing something permit anyone just not meet their expectations. For one, you have the right information that's generously succumbed this article for for you to look over before beginning your accomplish this.