  Cappell and Shaneson pointed out in 1978 interesting properties of Browder -
Livesay invariants which are similar to differentials in some spectral
sequence. Such spectral sequence was constructed in 1991 by Hambleton and
Kharshiladze. This spectral sequence is closely related to a problem of
realization of elements of Wall groups by normal maps of closed manifolds. The
main step of construction of the spectral sequence is an infinite filtration of
spectra in which only the first two, as is well-known, have clear geometric
sense. The first one is a spectrum $\Bbb L(\pi_1(X))$ for surgery obstruction
groups of a manifold $X$ and the second $\Bbb LP_*(F)$ is a spectrum for
surgery on a Browder-Livesay manifold pair $Y\subset X$. The geometric sense of
the third term of filtration was explained by Muranov, Repov\v{s}, and
Spaggiari in 2002. In the present paper we give a geometric interpretation of
all spectra of filtration in construction of Hambleton and Kharshiladze. We
introduce groups of obstructions to surgery on a system of embeddedd manifolds
and prove that spectra which realize these groups coincide with spectra in the
filtration of Hambleton and Kharshiladze. We describe algebraic and geometric
properties of introduced obstruction groups and their relations to the
classical surgery theory. We prove isomorphism between introduced groups and
Browder-Quinn $L$-groups of stratified manifolds. We give an application of our
results to closed manifold surgery problem and iterated Browder-Livesay
invariant.
