Article
KYUNGPOOK Math. J. 2019; 59(3): 563589
Published online September 23, 2019
Copyright © Kyungpook Mathematical Journal.
Note on the Codimension Two Splitting Problem
Yukio Matsumoto
Department of Mathematics, Gakushuin University, Mejiro, Toshimaku, Tokyo 1718588, Japan
email : yukiomat@math.gakushuin.ac.jp
Received: June 23, 2016; Accepted: November 11, 2016
Abstract
 Abstract
 Introduction
 Definitions and Statement of Results
 Surgery below the Middle Dimension
 Proof of Theorem 2.5 in the Odd Dimensional Case
 The Even Dimensional Case
 Some Properties of the Obstruction
 The Invariance of
γ (f ) underL equivalence ofM  Appendix
 Proposition A.1
 Remark A.2
 Acknowledgments and Postscript
 References
Let
Keywords: codimension two splitting problem, weak hregularity problem, codimension two surgery, surgery obstruction, relatively nonsingular Hermitian Ktheory.
Introduction
 Abstract
 Introduction
 Definitions and Statement of Results
 Surgery below the Middle Dimension
 Proof of Theorem 2.5 in the Odd Dimensional Case
 The Even Dimensional Case
 Some Properties of the Obstruction
 The Invariance of
γ (f ) underL equivalence ofM  Appendix
 Proposition A.1
 Remark A.2
 Acknowledgments and Postscript
 References
In this paper, we study
Relatively nonsingular Hermitian Ktheory is an interesting region of mathematics, but compared to the usual (nonsingular) Hermitian Ktheory^{2} (cf. [1, 25, 27, 31, 32]), it is still underdeveloped. Actually, after Ranicki’s remarkable work (see [28, §7.8 and §7.9], [29]), the relationship between our
Another interesting problem is to find a Künneth type formula for relatively nonsingular Hermitian Kgroups. The problem is to find the formula describing the Kgroups over
There should be close relationships between relatively nonsingular Hermitian Ktheory and algebraic number theory. In fact, Milnor’s and Levine’s papers [12, 24] seem to suggest certain connections of it to the class field theory.
Axiomatic foundations of relatively nonsingular Hermitian Kgroups are found in [3, 18, 28, 29]. See also [21].
For geometric applications of our theory, see [17, 19, 20, 22].
Recently there was remarkable progress related to [19] concerning spineless 4manifolds. Our example constructed in [19] was a compact PLspineless 4manifold homotopy equivalent to a 2torus
We remark here that M. H. Freedman [7] independently discovered the same Seifert forms as ours (see §5 and
The present paper is based on the author’s old note [16], which has been unpublished for more than forty years. The author hopes that the note would be still worth publishing, but an apology for such a long delay would be necessary. An explanation is given in
The (almost) verbatim reproduction of the old note [16] starts in the next paragraph after ∫ ∫ ∫. In the reproduction, we have updated the references^{3}. (In fact, in the old note, even the references [8] and [17] were cited as “to appear”. The papers [5], [6] were not available even in the preprint form. At that time, the only papers of Cappell and Shaneson that were available to the author were [4] in preprint form.) Also we have added some footnotes.
Now the reproduction starts.
In our previous paper [17], we introduced ambient surgery obstruction groups
The groups
Cappell and Shaneson [4] treated the same problem independently from homology surgery point of view. They state their obstruction in terms of Γgroups introduced by them. Naturally their Γgroups and our
Definitions and Statement of Results
 Abstract
 Introduction
 Definitions and Statement of Results
 Surgery below the Middle Dimension
 Proof of Theorem 2.5 in the Odd Dimensional Case
 The Even Dimensional Case
 Some Properties of the Obstruction
 The Invariance of
γ (f ) underL equivalence ofM  Appendix
 Proposition A.1
 Remark A.2
 Acknowledgments and Postscript
 References
Throughout the paper, we will work in the PLcategory^{4}. All manifolds are compact connected and oriented. All submanifolds are locally flat unless the contrary is stated. If a submanifold
The dimension of a manifold is indicated by a superscript.
Surgery below the Middle Dimension
 Abstract
 Introduction
 Definitions and Statement of Results
 Surgery below the Middle Dimension
 Proof of Theorem 2.5 in the Odd Dimensional Case
 The Even Dimensional Case
 Some Properties of the Obstruction
 The Invariance of
γ (f ) underL equivalence ofM  Appendix
 Proposition A.1
 Remark A.2
 Acknowledgments and Postscript
 References
Suppose the diagram (*) (on the previous page) is given. In this section we will perform surgery on
We will introduce the following notation:
ℱ
ℱ
Let
Lemma 3.1
Although the proof is not difficult, it is long and tedious, so we omit it. An analogous argument is done in [15].
To makeSuppose
Thus any element in
The effect of this surgery is to kill
Hereafter we will assume that
Since
The corresponding kernels are denoted by
Assume
If
Denote the restrictions of
The purpose of this paragraph is to prove the following:
The next lemma is useful in proving Theorem 3.3.
From the homotopy exact sequence of Φ, we have
This implies
Let
Let
Let
By
Here (1 −
Now note that
From this and the homotopy exact sequences of
Consider the following diagram obtained by MayerVietoris sequences:
Here we have used the hypothesis that
On the other hand, from the homology exact sequences it follows that
This and
We need an elementary algebraic lemma.
Lemma 3.5
Suppose there were a nonzero element
By Lemma 3.5 together with
Note that 0 ≌
This is the conclusion (
This is the conclusion (
We see that
Apply this to
Consider the exact sequence of Φ:
Cases where
First recall Namioka’s theorem [26]. We state it in our situation.
Theorem 3.6
(Namioka’s Theorem)
We now want to prove that our condition “
From Λhomology exact sequence of Φ, we have
Thus
Cases where
Inductively we assume
Perform ambient codimension 2 handle exchange along
Apply Namioka’s theorem (I) with
Proceeding inductively we will have
This completes the proof of Theorem 3.3.
Proof of Theorem 2.5 in the Odd Dimensional Case
 Abstract
 Introduction
 Definitions and Statement of Results
 Surgery below the Middle Dimension
 Proof of Theorem 2.5 in the Odd Dimensional Case
 The Even Dimensional Case
 Some Properties of the Obstruction
 The Invariance of
γ (f ) underL equivalence ofM  Appendix
 Proposition A.1
 Remark A.2
 Acknowledgments and Postscript
 References
Suppose
There are two exact sequences which contain
In §3 we performed surgery below the middle dimension, thus we may assume
Then from
By
Our task is to make
We will show that
First suppose
Moreover, it is easily verified that the attaching framed
Conversely, suppose that
The Even Dimensional Case
 Abstract
 Introduction
 Definitions and Statement of Results
 Surgery below the Middle Dimension
 Proof of Theorem 2.5 in the Odd Dimensional Case
 The Even Dimensional Case
 Some Properties of the Obstruction
 The Invariance of
γ (f ) underL equivalence ofM  Appendix
 Proposition A.1
 Remark A.2
 Acknowledgments and Postscript
 References
Suppose
In order to prove Theorem 2.5 in the even dimensional case, we first introduce the notion of
Let as before. Wall [32, Lemma 2.3] proved that
In order to state the properties we introduce some notations:
ℱ
ℱ
Theorem 5.1

f :W →V is homotopic (rel. ∂W )to a map (again denoted by f )which is tregular along ℱT with f ^{−1}(T ) =W ^{*}. 
f W ^{*} :W ^{*} →T is a simple homotopy equivalence. 
f ℱW ^{*} : ℱW ^{*} → ℱT is a Λ′homology equivalence. 
f U ^{*} :U ^{*} →F is a Λ′homology equivalence. 
π _{i}(E ^{*}, ℱN ^{*}) = 0for i ≦n ,and π _{n+1}(E ^{*}, ℱN ^{*})is a free Λmodule with the basis ∂ẽ _{1}, . . . ,∂ẽ _{r}.
Remark 5.2
Λ′homology of

This was done in the proof of “Fundamental Lemma” of [15, § 2.1]. We will state the result in our present situation; we proved there that if the core disks of
R _{i}’s (representing elements ofπ _{n+1}(E , ℱN )) are mapped to zero inπ _{n+1}(F , ℱT ) byf , then we can construct the desired homotopy which “splits” alongT .The condition is satisfied in our situation; the core disks of
R _{i}’s are of the form∂ẽ _{i},ẽ _{i} ∈π _{n+2}(Φ), and so they are mapped to zero inπ _{n+1}(F , ℱT ). This follows by the exactness of (α ). 
A proof was given in [17, Lemma 5.2]. We repeat it here for completeness.
Let
According to [32, Lemma 2.5], in order to prove that
The bases of
(iii) and (iv) follow easily from (ii) and the hypothesis that
(v) is obvious by the construction of
The proof of Theorem 5.1 is completed.
Because of property (ii) of
Lemma 5.3
Lemma 5.4
These lemmas will be proved in §5.2.
Seifert FormsLet us recall the definition of Seifert (−1)
A triple (
For any fixed
Λ′⊗_{Λ}
The following diagrams are commutative:
A free Seifert (−1)
(
([17]) The group
Note that (
Let
The following properties are important to our purpose:
Λ′⊗_{Λ}
Elements
We can now prove Lemma 5.3.
The obstruction
These
Let
As we proved in [8], the submanifold
Now consider the preimage
Let
Let
The effect of the surgery on
Now we consider the diagram (*) above again. Since we have made the vertical map on the right an isomorphism, we get the exact sequence (with Λcoefficients):
Note that the kernel of
Notice that Λ′⊗_{Λ}
Thus the Seifert form on
Now Theorem 2.5 follows immediately from Lemmas 5.3 and 5.4; we define
Some Properties of the Obstruction
 Abstract
 Introduction
 Definitions and Statement of Results
 Surgery below the Middle Dimension
 Proof of Theorem 2.5 in the Odd Dimensional Case
 The Even Dimensional Case
 Some Properties of the Obstruction
 The Invariance of
γ (f ) underL equivalence ofM  Appendix
 Proposition A.1
 Remark A.2
 Acknowledgments and Postscript
 References
6.1. Geometric Periodicity
Theorem 6.1
In the odd dimensional case, this is obvious by the definition of
The Invariance of γ (f ) under L equivalence of M
 Abstract
 Introduction
 Definitions and Statement of Results
 Surgery below the Middle Dimension
 Proof of Theorem 2.5 in the Odd Dimensional Case
 The Even Dimensional Case
 Some Properties of the Obstruction
 The Invariance of
γ (f ) underL equivalence ofM  Appendix
 Proposition A.1
 Remark A.2
 Acknowledgments and Postscript
 References
Let
Theorem 6.2
Corollary 6.3
Since
We will collect some known results on the structure of the group
Theorem 6.4

P _{2k+1}(1 →C →π →π′ → 1) =L _{2k+1}(π′ ) 
${P}_{m}\left(1\to 1\to \pi \stackrel{\text{id}}{\to}\pi \to 1\right)={L}_{m}(\pi )$ ,in particular P _{m}(1 → 1 → 1 → 1 → 1) =L _{m}(1), 
(
C _{m−1}is the knot cobordism group of (m − 1,m + 1)knots. )
The isomorphism (i) is the definition of
Let
Theorem 6.5
For the proof of Theorem 6.5, see [17].
Let ℰ denote an extension
Theorem 6.6
Corollary 6.7
According to Theorem 6.4 (iii), . Levine [11, 12] has proved that
By the periodicity of
It is proved in [17] that is onto and any element
Here we recall the main result of [17]: An oriented manifold pair (
Now returning to our present situation, we have
Suppose that the element
Let
Appendix
 Abstract
 Introduction
 Definitions and Statement of Results
 Surgery below the Middle Dimension
 Proof of Theorem 2.5 in the Odd Dimensional Case
 The Even Dimensional Case
 Some Properties of the Obstruction
 The Invariance of
γ (f ) underL equivalence ofM  Appendix
 Proposition A.1
 Remark A.2
 Acknowledgments and Postscript
 References
In [17] we show that a kind of intersection form can be defined on
Let
the composition
A nice immersion
Two nice immersions
Assume that two pathed nice immersions
Let {
An auxiliary pairing is defined by
In order to define the pairing
Let
Next we will define an element
Let
By the condition (iii) of nice immersions,
A.1. Orientation Conventions
For an oriented manifold
Proposition A.1
 Abstract
 Introduction
 Definitions and Statement of Results
 Surgery below the Middle Dimension
 Proof of Theorem 2.5 in the Odd Dimensional Case
 The Even Dimensional Case
 Some Properties of the Obstruction
 The Invariance of
γ (f ) underL equivalence ofM  Appendix
 Proposition A.1
 Remark A.2
 Acknowledgments and Postscript
 References
([17, Theorems 2.5, 2.9])
Thus we can define the following maps.
Remark A.2
 Abstract
 Introduction
 Definitions and Statement of Results
 Surgery below the Middle Dimension
 Proof of Theorem 2.5 in the Odd Dimensional Case
 The Even Dimensional Case
 Some Properties of the Obstruction
 The Invariance of
γ (f ) underL equivalence ofM  Appendix
 Proposition A.1
 Remark A.2
 Acknowledgments and Postscript
 References
Acknowledgments and Postscript
 Abstract
 Introduction
 Definitions and Statement of Results
 Surgery below the Middle Dimension
 Proof of Theorem 2.5 in the Odd Dimensional Case
 The Even Dimensional Case
 Some Properties of the Obstruction
 The Invariance of
γ (f ) underL equivalence ofM  Appendix
 Proposition A.1
 Remark A.2
 Acknowledgments and Postscript
 References
I forget exactly when I finished my old note [16], but seeing the bibliography of the note, I infer that it was not later than 1973. I submitted the note to
Since then I began considering the possibility of publishing the note somewhere. Akio Kawauchi encouraged me, and suggested Kyungpook Mathematical Journal having a long tradition in Korea as a possible journal to which my note should be submitted. The final impulse was given by Kokoro Tanaka and Reiko Shinjo, who kindly offered to convert my note (written on an oldfashioned typewriter) into a TeX file. I am very grateful to these friends of mine for their benevolent encouragement and cooperation, without which this paper would have never been published.
Finally, I deeply thank Professor Yongju Bae for his kind suggestion of the possibility of publishing my note in KMJ.
References
 Abstract
 Introduction
 Definitions and Statement of Results
 Surgery below the Middle Dimension
 Proof of Theorem 2.5 in the Odd Dimensional Case
 The Even Dimensional Case
 Some Properties of the Obstruction
 The Invariance of
γ (f ) underL equivalence ofM  Appendix
 Proposition A.1
 Remark A.2
 Acknowledgments and Postscript
 References
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