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The inertia subgroup $I_n(\pi)$ of a surgery obstruction group $L_n(\pi)$ isgenerated by elements which act trivially on the set of homotopy triangulations$\Cal S(X)$ for some closed topological manifold $X^{n-1}$ with $\pi_1(X)=\pi$.This group is a subgroup of the group $C_n(\pi)$ which consists of the elementswhich can be realized by normal maps of closed manifolds. In all known casesthese groups coincide and the computation of them is one of the basic problemsof surgery theory. The computation of the group $C_n(\pi)$ is equivalent to thecomputation the image of the assembly map $A:H_{n}(B\pi, \bold L_{\bullet})\toL_{n}(\pi)$. Every Browder-Livesay filtration of the manifold $X$ provides acollection of Browder-Livesay invariants which are the forbidden invariants inthe closed manifold surgery problem. In the present paper we describe allpossible forbidden invariants which can give a Browder-Livesay filtration forcomputing the inertia subgroup. Our approach is a natural generalization of theapproach of Hambleton and Kharshiladze. More precisely, we prove that aBrowder-Livesay filtration of a given manifold can give the following forbiddeninvariants for an element $x\in L_n(\pi_1(X))$ to belong to the subgroup$I_n(\pi)$: the nontrivial Browder-Livesay invariants in codimensions 0, 1, 2and a nontrivial class of obstructions of a restriction of a normal map to asubmanifold in codimension 3.

Surgery in codimension 3 and the Browder--Livesay invariants