Four-Dimensional Complexes with Fundamental Class Cavicchioli, Alberto; Hegenbarth, Friedrich; Spaggiari, Fulvia
Mediterranean journal of mathematics,
12/2020, Volume:
17, Issue:
6
Journal Article
Peer reviewed
This paper continues the study of 4-dimensional complexes from our previous work Cavicchioli et al. (Homol Homotopy Appl 18(2):267–281, 2016; Mediterr J Math 15(2):61, 2018.
...https://doi.org/10.1007/s00009-018-1102-3
) on the computation of Poincaré duality cobordism groups, and Cavicchioli et al. (Turk J Math 38:535–557, 2014) on the homotopy classification of strongly minimal
PD
4
-complexes. More precisely, we introduce a new class of oriented four-dimensional complexes which have a “fundamental class”, but do not satisfy Poincaré duality in all dimensions. Such complexes with partial Poincaré duality properties, which we call
SFC
4
-complexes, are very interesting to study and can be classified, up to homotopy type. For this, we introduce the concept of resolution, which allows us to state a condition for a
SFC
4
-complex to be a
PD
4
-complex. Finally, we obtain a partial classification of
SFC
4
-complexes. A future goal will be a classification in terms of algebraic
SFC
4
-complexes similar to the very satisfactory classification result of
PD
4
-complexes obtained by Baues and Bleile (Algebraic Geom. Topol. 8:2355–2389, 2008).
We apply the Gromov–Hausdorff metric
d
G
for characterization of certain generalized manifolds. Previously, we have proven that with respect to the metric
d
G
,
generalized
n
-manifolds are limits of ...spaces which are obtained by gluing two topological
n
-manifolds by a controlled homotopy equivalence (the so-called 2-patch spaces). In the present paper, we consider the so-called
manifold-like
generalized
n
-manifolds
X
n
,
introduced in 1966 by Mardeić and Segal, which are characterized by the existence of
δ
-mappings
f
δ
of
X
n
onto closed manifolds
M
δ
n
,
for arbitrary small
δ
>
0
, i.e., there exist onto maps
f
δ
:
X
n
→
M
δ
n
such that for every
u
∈
M
δ
n
,
f
δ
-
1
(
u
)
has diameter less than
δ
. We prove that with respect to the metric
d
G
,
manifold-like generalized
n
-manifolds
X
n
are limits of topological
n
-manifolds
M
i
n
. Moreover, if topological
n
-manifolds
M
i
n
satisfy a certain local contractibility condition
M
(
ϱ
,
n
)
, we prove that generalized
n
-manifold
X
n
is resolvable.
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 $X^{n}$ be an oriented closed generalized $n$-manifold, $n\ge 5$. In our recent paper (Proc. Edinb. Math. Soc. (2) 63 (2020), no. 2, 597–607), we have constructed a map $t:\mathcal {N}(X^{n}) \to ...H^{st}_{n} ( X^{n}; \mathbb{L}^{+})$ which extends the normal invariant map for the case when $X^{n}$ is a topological $n$-manifold. Here, $\mathcal {N}(X^{n})$ denotes the set of all normal bordism classes of degree one normal maps $(f,\,b): M^{n} \to X^{n},$ and $H^{st}_{*} ( X^{n}; \mathbb{E})$ denotes the Steenrod homology of the spectrum $\mathbb{E}$. An important non-trivial question arose whether the map $t$ is bijective (note that this holds in the case when $X^{n}$ is a topological $n$-manifold). It is the purpose of this paper to prove that the answer to this question is affirmative.
In this paper we continue our investigations of 4-dimensional complexes in A. Cavicchioli, F. Hegenbarth, F. Spaggiari,
, Mediterr. J. Math.
(2020), 175. We study a class of finite oriented ...4-complexes which we call
-complexes, defined as follows. An
is a 4-dimensional finite oriented
-complex
with a single 4-cell such that
with a fundamental class
. By well-known results of Wall, any Poincaré complex is of this type. We are interested in two questions. First, for which 3-complexes
does an element
exist such that
is a Poincaré complex? Second, if there exists one, how many others can be constructed from
? The latter question was addressed studied in the above cited previous paper of the authors. In the present paper we deal with the first problem, and give necessary and sufficient conditions on
and
to satisfy Poincaré duality with
- and Λ-coefficients. Here Λ denotes the integral group ring of
. Before, we give a classification of all
-complexes based on the finite 3-complex
, and make some remarks concerning
- and Λ-Poincaré duality.
Abstract
In this paper we continue our investigations of 4-dimensional complexes in A. Cavicchioli, F. Hegenbarth, F. Spaggiari,
Four-dimensional complexes with fundamental class
, Mediterr. J. Math.
...17
(2020), 175. We study a class of finite oriented 4-complexes which we call
FC
4
{\mathrm{FC}_{4}}
-complexes, defined as follows. An
FC
4
{\mathrm{FC}_{4}}
-
complex
is a 4-dimensional finite oriented
CW
-complex
X
with a single 4-cell such that
H
4
(
X
,
ℤ
)
≅
ℤ
{H_{4}(X,\mathbb{Z})\cong\mathbb{Z}}
with a fundamental class
X
∈
H
4
(
X
,
ℤ
)
{X\in H_{4}(X,\mathbb{Z})}
. By well-known results of Wall, any Poincaré complex is of this type. We are interested in two questions. First, for which 3-complexes
K
does an element
φ
∈
π
3
(
K
)
{\varphi\in\pi_{3}(K)}
exist such that
K
∪
φ
D
4
{K\cup_{\varphi}D^{4}}
is a Poincaré complex? Second, if there exists one, how many others can be constructed from
K
? The latter question was addressed studied in the above cited previous paper of the authors. In the present paper we deal with the first problem, and give necessary and sufficient conditions on
K
and
φ
∈
π
3
(
K
)
{\varphi\in\pi_{3}(K)}
to satisfy Poincaré duality with
ℤ
{\mathbb{Z}}
- and Λ-coefficients. Here Λ denotes the integral group ring of
π
1
(
K
)
{\pi_{1}(K)}
. Before, we give a classification of all
FC
4
{\mathrm{FC}_{4}}
-complexes based on the finite 3-complex
K
, and make some remarks concerning
ℤ
{\mathbb{Z}}
- and Λ-Poincaré duality.
Abstract
The aim of this paper is to show the importance of the Steenrod construction of homology theories for the disassembly process in surgery on a
generalized n
-manifold
X
n
, in order to ...produce an element of generalized homology theory, which is basic for calculations. In particular, we show how to construct an element of the
n
th Steenrod homology group
$H^{st}_{n} (X^{n}, \mathbb {L}^+)$
, where
+
is the connected covering spectrum of the periodic surgery spectrum , avoiding the use of the
geometric
splitting procedure, the use of which is standard in surgery on
topological
manifolds.
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.
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