Surgery Obstruction
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, specifically in surgery theory, the surgery obstructions define a map theta colon mathcal{N} (X) to L_n (pi_1 (X)) from the normal invariants to the L-groups which is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property when n geq 5:A degree one normal map (f,b) colon M to X is normally cobordant to a homotopy equivalence if and only if the image (f,b) = 0 in L_n (mathbb{Z} [pi_1 (X)])If the…mehr

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, specifically in surgery theory, the surgery obstructions define a map theta colon mathcal{N} (X) to L_n (pi_1 (X)) from the normal invariants to the L-groups which is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property when n geq 5:A degree one normal map (f,b) colon M to X is normally cobordant to a homotopy equivalence if and only if the image (f,b) = 0 in L_n (mathbb{Z} [pi_1 (X)])If the element (f,b) is zero in the L-group surgery can be done on M to modify f to a homotopy equivalence.Geometrically the reason why this is not always possible is that performing surgery in the middle dimension to kill an element in K_k (tilde M) possibly creates an element in K_{k-1} (tilde M) when n = 2k or in K_{k} (tilde M) when n = 2k + 1. So this possibly destroys what has already been achieved. However, if (f,b) is zero, surgeries can be arranged in such a way that this does not happen.