** Authors:**
FRIEDRICH HEGENBARTH, YURI MURANOV, DUSAN REPOVS

** Abstract: **
The inertia subgroup I_n(\pi) of a surgery obstruction group
L_n(\pi) is generated by elements that act trivially on the set of
homotopy triangulations 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 elements that can be
realized by normal maps of closed manifolds. These 2 groups coincide
by a recent result of Hambleton, at least for n \geq 6 and in all
known cases. In this paper we introduce a subgroup J_n(\pi) \subset
I_n(\pi), which is generated by elements of the group L_n(\pi),
which act trivially on the set S^{\partial}(X, \partial X) of
homotopy triangulations relative to the boundary of any compact
manifold with boundary (X, \partial X). Every Browder--Livesay
filtration of the manifold X provides a collection of higher-order
Browder--Livesay invariants for any element x \in L_n(\pi). In the
present paper we describe all possible invariants that can give a
Browder--Livesay filtration for computing the subgroup J_n(\pi).
These are invariants of elements x \in L_n(\pi), which are nonzero if
x \notin J_n(\pi). More precisely, we prove that a Browder--Livesay
filtration of a given manifold can give the following invariants of
elements x \in L_n(\pi), which are nonzero if x \notin J_n(\pi): the
Browder-Livesay invariants in codimensions 0, 1, 2 and a class of
obstructions of the restriction of a normal map to a submanifold in
codimension 3.

** Keywords: **
Surgery assembly map, closed manifolds surgery problem, assembly map,
inertia subgroup, splitting problem, Browder--Livesay invariants,
Browder--Livesay groups, normal maps, iterated Browder--Livesay
invariants, manifold with filtration, Browder--Quinn surgery
obstruction groups, elements of the second type of a Wall group

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