Controlled Surgery and L-Homology Hegenbarth, Friedrich; Repovš, Dušan
Mediterranean journal of mathematics,
06/2019, Volume:
16, Issue:
3
Journal Article
Peer reviewed
Open access
This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map
(
f
,
b
)
:
M
n
→
X
n
with control map
q
:
X
n
→
B
to complete controlled ...surgery is an element
σ
c
(
f
,
b
)
∈
H
n
(
B
,
L
)
, where
M
n
,
X
n
are topological manifolds of dimension
n
≥
5
. Our proof uses essentially the geometrically defined
L
-spectrum as described by Nicas (going back to Quinn) and some well-known homotopy theory. We also outline the construction of the algebraically defined obstruction, and we explicitly describe the assembly map
H
n
(
B
,
L
)
→
L
n
(
π
1
(
B
)
)
in terms of forms in the case
n
≡
0
(
4
)
. Finally, we explicitly determine the canonical map
H
n
(
B
,
L
)
→
H
n
(
B
,
L
0
)
.
The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincaré duality complexes (PD complexes). The problem is that an arbitrary generalized manifold X ...is always an ENR space, but it is not necessarily a complex. Moreover, finite PD complexes require the Poincaré duality with coefficients in the group ring Λ (Λ-complexes). Standard homology theory implies that X is a Z-PD complex. Therefore by Browder's theorem, X has a Spivak normal fibration which in turn, determines a Thom class of the pair (N,∂N) of a mapping cylinder neighborhood of X in some Euclidean space. Then X satisfies the Λ-Poincaré duality if this class induces an isomorphism with Λ-coefficients. Unfortunately, the proof of Browder's theorem gives only isomorphisms with Z-coefficients. It is also not very helpful that X is homotopy equivalent to a finite complex K, because K is not automatically a Λ-PD complex. Therefore it is convenient to introduce Λ-PD structures. To prove their existence on X, we use the construction of 2-patch spaces and some fundamental results of Bryant, Ferry, Mio, and Weinberger. Since the class of all Λ-PD complexes does not contain all generalized manifolds, we appropriately enlarge this class and then describe (i.e. recognize) generalized manifolds within this enlarged class in terms of the Gromov–Hausdorff metric.
We consider an open question: Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence? The answer is known to be positive for 1-dimensional polyhedra and ...polyhedra with virtually-polycyclic fundamental groups, so it is natural to ask about 2-dimensional polyhedra, in particular about those with solvable fundamental groups.
In this paper we prove that for each 2-dimensional polyhedron P with weakly Hopfian fundamental group, every homotopy domination of P over itself is a homotopy equivalence. A group is weakly Hopfian if it is not isomorphic to a proper retract of itself. Thus every Hopfian group is weakly Hopfian.
The class of Hopfian groups contains: all torsion-free hyperbolic groups, finitely generated linear groups, knot groups, limit groups, and many others.
One corollary to the main result is that for 2-dimensional polyhedra with elementary amenable (including virtually-solvable) fundamental groups of finite cohomological dimension, the answer to our question is positive (we show that every elementary amenable group with finite cohomological dimension is Hopfian).
The problem in consideration is related in an obvious way to the famous question of K. Borsuk (1967): Is it true that two compact ANR's homotopy dominating each other have the same homotopy type?
is the connected covering spectrum of \mathbb{L} various stages of the Postnikov tower of , one obtains an interesting filtration of the controlled structure set.>
Let $G$ be a finite group. For a certain class of CW-complexes with a $G$-action which are equivariantly dominated by a finite complex we define algebraic invariants to decide when the space is ...equivariantly homotopy or homology equivalent to a finite complex.
It is proved that a fibered compact metric space having the shape of a CW complex has the homotopy type of that complex, and that its Wall obstruction to finiteness is zero.
This paper demonstrates the effects on local mass transfer of placing a variety of partial obstructions in the form of fences or steps (arranged in single or multiple configurations) on the walls of ...a parallel plate electrochemical flow cell. For the wall on which the fence is placed, a plot of the position of mass transfer relative to an obstruction against the Reynolds number for different fence heights shows that, for each fence, the distance to peak position initially increases with the Reynolds number, but tends to decrease at higher Reynolds numbers. For the wall opposite to the fence, a pronounced peak immediately opposite the obstruction, corresponding to a position of maximum velocity, and a second downstream peak, corresponding to a recirculation/reattachment zone, were identified. A correlation of the peak Sherwood number as a function of the Peclet number for all fence heights and also for a backward-facing step was compared with data of other workers. The numerical prediction of flow reattachment to the walls produced excellent agreement with the positions of peak mass transfer for both the fences and the step, but agreed less well in terms of magnitude.
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