arXiv subjects

Friedrich Hegenbarth

Publications and source records attributed to Friedrich Hegenbarth.

15 recordsLinked to original sources

arXiv2025

On geometric representation of $\mathbb{L}$-homology classes

In this chapter we give a geometric representation of $H_{n}(B;\mathbb{L})$ classes, where $\mathbb{L}$ is the $4$-periodic surgery spectrum, by establishing a relationship between the normal cobordism classes ${\mathcal{N}}^{H}_{n}(B,\partial)$ and the $n$-th $\mathbb{L}$-homology of $B$, representing the elements of $H_{n}(B;\mathbb{L})$ by normal degree one maps with a reference map to $B$. More precisely, we prove that for every $n \ge 6$ and every finite complex $B,$ there exists a map $Γ: H_n(B;\mathbb{L}) \longrightarrow \mathcal{N}^{H}_{n}(B,\partial).$

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arXiv2023

Limits of manifolds in the Gromov-Hausdorff metric space

We apply the Gromov-Hausdorff metric $d_G$ for characterization of certain generalized manifolds. Previously, we have proved 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 {\sl manifold-like} generalized $n$-manifolds $X^{n},$ introduced in 1966 by Mardešić and Segal, which are characterized by the existence of $δ$-mappings $f_δ$ of $X^n$ onto closed manifolds $M^{n}_δ,$ for arbitrary small $δ>0$, i.e. there exist onto maps $f_δ\colon X^{n}\to M^{n}_δ$ such that for every $u\in M^{n}_δ$, $f^{-1}_δ(u)$ 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 $M^{n}_{i}$. Moreover, if topological $n$-manifolds $M^{n}_{i}$ satisfy a certain local contractibility condition $\mathcal{M}(\varrho, n)$, we prove that generalized $n$-manifold $X^{n}$ is resolvable.

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arXiv2022

Generalized manifolds, normal invariants, and $\mathbb{L}$-homology

Let $X^{n}$ be an arbitrary 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 nontrivial question arose whether the map $t$ is bijective (note that this holds in the case that $X^{n}$ is a topological $n$-manifold). It is the purpose of this paper to prove that the answer to this question is affirmative.

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arXiv2020

Controlled surgery and $\mathbb{L}$-homology

This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map $(f,b): M^n \rightarrow X^n$ with control map $q: X^n \rightarrow B$ to complete controlled surgery is an element $σ^c (f, b) \in H_n (B, \mathbb{L})$, where $M^n, X^n$ are topological manifolds of dimension $n \geq 5$. Our proof uses essentially the geometrically defined $\mathbb{L}$-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 $H_n (B, \mathbb{L}) \rightarrow L_n (π_1 (B))$ in terms of forms in the case $n \equiv 0 (4)$. Finally, we explicitly determine the canonical map $H_n (B, \mathbb{L}) \rightarrow H_n (B, L_0)$.

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arXiv2020

Controlled homotopy equivalences and structure sets of manifolds

For a closed topological $n$--manifold $K$ and a map $p:K\to B$ inducing an isomorphism $π_1(K)\toπ_1(B)$, there is a canonicaly defined morphism $b:H_{n+1}(B,K,\mathbb{L})\to \mathbb{S} (K)$, where $\mathbb{L}$ is the periodic simply-connected surgery spectrum and $\mathbb{S} (K)$ is the topological structure set. We construct a refinement $a:H_{n+1}^{+}(B,K,\mathbb{L} )\to \mathbb{S}_{\varepsilon ,δ}(K)$ in the case when $p$ is $UV^1$, and we show that $a$ is bijective if $B$ is a finite-dimensional compact metric ANR. Here, $H_{n+1}^{+}(B,K,\mathbb{L} )\subset H_{n+1}(B,K,\mathbb{L} )$, and $\mathbb{S}_{\varepsilon ,δ}(K)$ is the controlled structure set. We show that the Pedersen-Quinn-Ranicki controlled surgery sequence is equivalent to the exact $\mathbb{L}$-homology sequence of the map $p:K \to B$, i.e. that $$H_{n+1}(B,\mathbb{L})\to H_{n+1}^{+}(B,K,\mathbb{L} )\to H_n(K,\mathbb{L}^{+})\to H_n(B,\mathbb{L} ), \ \mathbb{L}^{+}\to \mathbb{L},$$ is the connected covering spectrum of $\mathbb{L}$. By taking for $B$ various stages of the Postnikov tower of $K$, one obtains an interesting filtration of the controlled structure set.

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arXiv2020

On Steenrod $\mathbb{L}$-homology, generalized manifolds, and surgery

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 $\mathbb{L}^+$ is the connected covering spectrum of the periodic surgery spectrum $\mathbb{L}$, avoiding the use of the geometric splitting procedure, which is standardly used in surgery on topological manifolds.

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arXiv2019

s-Cobordism classification of $4$-manifolds through the group of homotopy self-equivalences

The aim of this paper is to give an $s$-cobordism classification of topological $4$-manifolds in terms of the standard invariants using the group of homotopy self-equivalences. Hambleton and Kreck constructed a braid to study the group of homotopy self-equivalences of $4$-manifolds. Using this braid together with the modified surgery theory of Kreck, we give an $s$-cobordism classification for certain $4$-manifolds with fundamental group $π$, such that cd $π\leq 2$.

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arXiv2018

The relationship of generalized manifolds to Poincaré duality complexes and topological manifolds

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 $\mathbb{Z}$-PD complex. Therefore by Browder's theorem, $X$ has a Spivak normal fibration which in turn, determines a Thom class of the pair $(N,\partial N)$ 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 $\mathbb{Z}$-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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arXiv2015

Homotopy classification of $PD_4$-complexes relative an order relation

We define an order relation among oriented $PD_4$-complexes. We show that with respect to this relation, two $PD_4$-complexes over the same complex are homotopy equivalent if and only if there is an isometry between the second homology groups. We also consider minimal objects of this relation.

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arXiv2014

On minimal Poincaré $4$-complexes

We consider two types of minimal Poincaré $4$-complexes. One is defined with respect to the degree $1$-map order. This idea was already present in our previous papers, and more systematically studied later by Hillman. The second type of minimal Poincaré $4$-complexes were introduced by Hambleton, Kreck and Teichner. It is not based on an order relation. In the present paper we study existence and uniqueness.

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arXiv2013

Browder-Livesay filtrations and the example of Cappell and Shaneson

Let $M^3$ be a 3-dimensional manifold with fundamental group $π_1(M)$ which contains a quaternion subgroup $Q$ of order 8. In 1979 Cappell and Shaneson constructed a nontrivial normal map $ f\colon M^3\times T^2\to M^3\times S^2$ which cannot be detected by simply connected surgery obstructions along submanifolds of codimension 0, 1, or 2, but it can be detected by the codimension 3 Kervaire-Arf invariant. The proof of non-triviality of $σ(f)\in L_5(π_1(M))$ is based on consideration of a Browder-Livesay filtration of a manifold $X$ with $π_1(X)\cong π_1(M)$. For a Browder-Livesay pair $Y^{n-1}\subset X^n$, the restriction of a normal map to the submanifold $Y$ is given by a partial multivalued map $Γ\colon L_n(π_1(X))\to L_{n-1}(π_1(Y))$, and the Browder-Livesay filtration provides an iteration $Γ^n$. This map is a basic step in the definition of the iterated Browder-Livesay invariants which give obstructions to realization of surgery obstructions by normal maps of closed manifolds. In the present paper we prove that $Γ^3(σ(f))=0$ for any Browder-Livesay filtration of a manifold $X^{4k+1}$ with $π_1(X)\cong Q$. We compute splitting obstruction groups for various inclusions $ρ\to Q$ of index 2, describe natural maps in the braids of exact sequences, and make more precise several results about surgery obstruction groups of the group $Q$.

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arXiv2011

The Bryant-Ferry-Mio-Weinberger construction of generalized manifolds

Following Bryant, Ferry, Mio and Weinberger we construct generalized manifolds as limits of controlled sequences p_i: X_i --> X_{i-1} : i = 1,2,... of controlled Poincaré spaces. The basic ingredient is the epsilon-delta-surgery sequence recently proved by Pedersen, Quinn and Ranicki. Since one has to apply it not only in cases when the target is a manifold, but a controlled Poincaré complex, we explain this issue very roughly. Specifically, it is applied in the inductive step to construct the desired controlled homotopy equivalence p_{i+1}: X_{i+1} --> X_i. Our main theorem requires a sufficiently controlled Poincaré structure on X_i (over X_{i-1}). Our construction shows that this can be achieved. In fact, the Poincaré structure of X_i depends upon a homotopy equivalence used to glue two manifold pieces together (the rest is surgery theory leaving unaltered the Poincaré structure). It follows from the epsilon-delta-surgery sequence (more precisely from the Wall realization part) that this homotopy equivalence is sufficiently well controlled. In the final section we give additional explanation why the limit space of the X_i's has no resolution.

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arXiv2008

Surgery in codimension 3 and the Browder--Livesay invariants

The inertia subgroup $I_n(π)$ of a surgery obstruction group $L_n(π)$ is generated by elements which act trivially on the set of homotopy triangulations $\Cal S(X)$ for some closed topological manifold $X^{n-1}$ with $π_1(X)=π$. This group is a subgroup of the group $C_n(π)$ which consists of the elements which can be realized by normal maps of closed manifolds. In all known cases these groups coincide and the computation of them is one of the basic problems of surgery theory. The computation of the group $C_n(π)$ is equivalent to the computation the image of the assembly map $A:H_{n}(Bπ, \bold L_{\bullet})\to L_{n}(π)$. Every Browder-Livesay filtration of the manifold $X$ provides a collection of Browder-Livesay invariants which are the forbidden invariants in the closed manifold surgery problem. In the present paper we describe all possible forbidden invariants which can give a Browder-Livesay filtration for computing the inertia subgroup. Our approach is a natural generalization of the approach of Hambleton and Kharshiladze. More precisely, we prove that a Browder-Livesay filtration of a given manifold can give the following forbidden invariants for an element $x\in L_n(π_1(X))$ to belong to the subgroup $I_n(π)$: the nontrivial Browder-Livesay invariants in codimensions 0, 1, 2 and a nontrivial class of obstructions of a restriction of a normal map to a submanifold in codimension 3.

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arXiv2006

Some recent approaches in 4-dimensional surgery theory

It is well-known that an n-dimensional Poincaré complex $X^n$, $n \ge 5$, has the homotopy type of a compact topological $n$-manifold if the total surgery obstruction $s(X^n)$ vanishes. The present paper discusses recent attempts to prove analogous result in dimension 4. We begin by reviewing the necessary algebraic and controlled surgery theory. Next, we discuss the key idea of Quinn's approach. Finally, we present some cases of special fundamental groups, due to the authors and to Yamasaki.

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arXiv2006

Applications of controlled surgery in dimension 4: Examples

The validity of Freedman's disk theorem is known to depend only on the fundamental group. It was conjectured that it fails for nonabelian free fundamental groups. If this were true then surgery theory would work in dimension four. Recently, Krushkal and Lee proved a surprising result that surgery theory works for a large special class of 4-manifolds with free nonabelian fundamental groups. The goal of this paper is to show that this also holds for other fundamental groups which are not known to be good, and that it is best understood using controlled surgery theory of Pedersen--Quinn--Ranicki. We consider some examples of 4-manifolds which have the fundamental group either of a closed aspherical surface or of a 3-dimensional knot space. A more general theorem is stated in the appendix.

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