KYUNGPOOK Math. J. 2019; 59(3): 563-589  
Note on the Codimension Two Splitting Problem
Yukio Matsumoto
Department of Mathematics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171-8588, Japan
e-mail : yukiomat@math.gakushuin.ac.jp
Received: June 23, 2016; Accepted: November 11, 2016; Published online: September 23, 2019.
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Abstract

Let W and V be manifolds of dimension m + 2, M a locally flat submanifold of V whose dimension is m. Let f : WV be a homotopy equivalence. The problem we study in this paper is the following: When is f homotopic to another homotopy equivalence g : WV such that g is transverse regular along M and such that g|g−1(M) : g−1(M) → M is a simple homotopy equivalence? López de Medrano (1970) called this problem the weak h-regularity problem. We solve this problem applying the codimension two surgery theory developed by the author (1973). We will work in higher dimensions, assuming that m ≧ 5.

Keywords: codimension two splitting problem, weak h-regularity problem, codimension two surgery, surgery obstruction, relatively non-singular Hermitian K-theory.
Introduction
roduction

In this paper, we study the weak h-regularity problem in the sense of López de Medrano [14], or the codimension two splitting problem, whose precise formulation will be given in §2. The main results will clarify the role of relatively non-singular Hermitian K-groups [3, 18] in the codimension two splitting problem. This type of K-groups (called P -groups in our theory) are defined over a surjective ring homo-morphism between (not necessarily commutative) rings ABand are, roughly speaking, the Witt groups of relatively non-singular Hermitian forms over AB, meaning that they are defined over A and become non-singular over B. For example, P-groups over are isomorphic1 to Levine’s knot cobordism groups [11]. See [17]. Cappell and Shaneson [5, 6], defined similar but slightly different Hermitian K-groups (called Γ-groups by them), and using these groups they studied the codimension two placement problems extensively. However, in the author’s opinion, our formulation is simpler than theirs. For example, we can give a unique element in our P -group as the obstruction to codimension two splitting (see Theorem 2.5 below), while in Cappell and Shaneson’s theory, the obstruction to the same splitting problem is given in two steps involving two different (but mutually related) groups (see [4], [5, §8], [6, p.438]).

Relatively non-singular Hermitian K-theory is an interesting region of mathematics, but compared to the usual (non-singular) Hermitian K-theory2 (cf. [1, 25, 27, 31, 32]), it is still under-developed. Actually, after Ranicki’s remarkable work (see [28, §7.8 and §7.9], [29]), the relationship between our P-groups and Cappell-Shaneson’s Γ-groups is still unclear. In Kato’s problem list [9], C. T. C. Wall proposed the problem of computing these groups [9, Problem 6.2, p.426].

Another interesting problem is to find a Künneth type formula for relatively non-singular Hermitian K-groups. The problem is to find the formula describing the K-groups over A ⊗ Λ → B ⊗ Λ in terms of K-groups over AB, where . It is known that Shaneson’s type of Künneth formula [30] cannot be expected here. See [19].

There should be close relationships between relatively non-singular Hermitian K-theory 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 non-singular Hermitian K-groups 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 4-manifolds. Our example constructed in [19] was a compact PL-spineless 4-manifold homotopy equivalent to a 2-torus T2. In 2018, using Heegaad Floer d invariants, A. S. Levine and T. Lidman [13] constructed compact PL-spineless 4-manifolds homotopy equivalent to a 2-sphere S2, and in 2019, H. J. Kim and D. Ruberman [10] proved that some of the Levine-Lidman manifolds admit tame topological spines.

We remark here that M. H. Freedman [7] independently discovered the same Seifert forms as ours (see §5 and Appendix of the present paper). Applying his codimension two surgery theory, he found higher dimensional counterexamples to the generalized Thom conjecture concerning the Betti numbers of smooth hypersurfaces in the complex projective spaces [7].

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 Acknowledgments and Postscript at the end of this paper.

The (almost) verbatim reproduction of the old note [16] starts in the next paragraph after ∫ ∫ ∫. In the reproduction, we have updated the references3. (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 Pm(ℰ) in codimension two. There we introduced them in order to describe the obstruction to finding locally flat spines of (m + 2)-manifolds which are simple homotopy equivalent to a Poincaré complex of formal dimension m.

The groups Pm(ℰ) work, however, as the obstruction groups for the weak h-regularity problem in the sense of López de Medrano [14] as well (i.e. the splitting problem in codimension two). The purpose of this note is to give a detailed proof of it.

Cappell and Shaneson [4] treated the same problem independently from homology surgery point of view. They state their obstruction in terms of Γ-groups introduced by them. Naturally their Γ-groups and our P-groups seem to be related to each other very closely. The relationship will be discussed elsewhere.

Definitions and Statement of Results
f Results

Throughout the paper, we will work in the PL-category4. All manifolds are compact connected and oriented. All submanifolds are locally flat unless the contrary is stated. If a submanifold M of W has a boundary, we always assume that it is properly embedded, i.e., j−1(∂W) = ∂M, where j : MW is the inclusion.

The dimension of a manifold is indicated by a superscript.

<div><strong>Definition 2.1</strong><p>([<span class="xref"><a href="#B8">8</a></span>, <span class="xref"><a href="#B17">17</a></span>]) Let <italic>M<sup>m</sup></italic> be a submanifold of <italic>W</italic><sup><italic>m</italic>+2</sup> with a regular neighborhood <italic>N</italic>. (If <italic>∂M</italic> ≠ ∅, <italic>N</italic> is assumed so that <italic>N</italic> ∩ <italic>∂W</italic> is a regular neighborhood of <italic>∂M</italic> in <italic>∂W</italic>.) The closed complement <inline-formula><math id="m3" display="inline"><mrow><mi>E</mi><mo>=</mo><mover accent='true'><mrow><mi>W</mi><mo>−</mo><mi>N</mi></mrow><mo>¯</mo></mover></mrow></math></inline-formula> is called the <italic>exterior</italic>, <italic>E</italic> ∩ <italic>N</italic> is called the <italic>frontier</italic> of <italic>N</italic> and is denoted by ℱ<italic>N</italic>. If <italic>π</italic><sub><italic>i</italic></sub>(<italic>E</italic>, ℱ<italic>N</italic>) = 0 for <italic>i</italic> ≦ <italic>k</italic>, <italic>M<sup>m</sup></italic> is said to be <italic>exterior k-connected</italic>.</p><p>Suppose <italic>M<sup>m</sup></italic> is exterior 2-connected in <italic>W</italic><sup><italic>m</italic>+2</sup>, then we have isomorphisms <inline-formula><math id="m4" display="inline"><mrow><msub><mrow><mi>π</mi></mrow><mn>1</mn></msub><mo stretchy='false'>(</mo><mi>ℱ</mi><mi>N</mi><mo stretchy='false'>)</mo><mover><mo stretchy='true'>→</mo><mo>≅</mo></mover><msub><mrow><mi>π</mi></mrow><mn>1</mn></msub><mo stretchy='false'>(</mo><mi>E</mi><mo stretchy='false'>)</mo></mrow></math></inline-formula> and <inline-formula><math id="m5" display="inline"><mrow><msub><mrow><mi>π</mi></mrow><mn>1</mn></msub><mo stretchy='false'>(</mo><mi>M</mi><mo stretchy='false'>)</mo><mover><mo stretchy='true'>→</mo><mo>≅</mo></mover><msub><mrow><mi>π</mi></mrow><mn>1</mn></msub><mo stretchy='false'>(</mo><mi>N</mi><mo stretchy='false'>)</mo><mover><mo stretchy='true'>→</mo><mo>≅</mo></mover><msub><mrow><mi>π</mi></mrow><mn>1</mn></msub><mo stretchy='false'>(</mo><mi>W</mi><mo stretchy='false'>)</mo></mrow></math></inline-formula>. The proof is not difficult. (See [<span class="xref"><a href="#B17">17</a></span>, Lemma 1.3].) The former group is denoted by <italic>π</italic>, and the latter by <italic>π′</italic>. Since ℱ <italic>N</italic> → <italic>M</italic> is an <italic>S</italic><sup>1</sup>-bundle, there is an exact sequence: <disp-formula id="FD3-kmj-59-563"><math id="m6" display='block'><mrow><msub><mrow><mi>π</mi></mrow><mn>2</mn></msub><mo stretchy='false'>(</mo><mi>M</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mrow><mi>π</mi></mrow><mn>1</mn></msub><mo stretchy='false'>(</mo><msup><mrow><mi>S</mi></mrow><mn>1</mn></msup><mo stretchy='false'>)</mo><mo>→</mo><msub><mrow><mi>π</mi></mrow><mn>1</mn></msub><mo stretchy='false'>(</mo><mi>ℱ</mi><mi>N</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mrow><mi>π</mi></mrow><mn>1</mn></msub><mo stretchy='false'>(</mo><mi>M</mi><mo stretchy='false'>)</mo><mo>→</mo><mn>1</mn><mo>.</mo></mrow></math></disp-formula>From this, we have the exact sequence <disp-formula id="FD4-kmj-59-563"><math id="m7" display='block'><mrow><mn>1</mn><mo>→</mo><mi>C</mi><mo>→</mo><mi>π</mi><mo>→</mo><mi>π</mi><mo>′</mo><mo>→</mo><mn>1</mn><mo>,</mo></mrow></math></disp-formula>where <italic>C</italic> = Coker(<italic>π</italic><sub>2</sub>(<italic>M</italic>) → <italic>π</italic><sub>1</sub>(<italic>S</italic><sup>1</sup>)), and <italic>C</italic> is a cyclic group in the center of <italic>π</italic> [<span class="xref"><a href="#B17">17</a></span>, Lemma 1.4]. We can specify a generator <italic>t</italic> of <italic>C</italic>. The generator <italic>t</italic> is canonically determined by the orientations of <italic>M<sup>m</sup></italic> and <italic>W</italic><sup><italic>m</italic>+2</sup> [<span class="xref"><a href="#B17">17</a></span>, Lemma 1.4].</p></div><div><strong>Definition 2.2</strong><p>([<span class="xref"><a href="#B17">17</a></span>]) The above exact sequence is denoted by <italic>ℰ</italic>, and is said to <italic>be associated with the exterior</italic> 2<italic>-connected manifold pair</italic> (<italic>W</italic><sup><italic>m</italic>+2</sup>, <italic>M<sup>m</sup></italic>).</p><p>Let <italic>P</italic>, <italic>Q</italic> be two manifolds, <italic>K</italic> a submanifold of <italic>Q</italic>, Let <italic>h</italic> : <italic>P</italic> → <italic>Q</italic> be a continuous map.</p></div><div><strong>Definition 2.3</strong><p>([<span class="xref"><a href="#B14">14</a></span>]) The map <italic>h is weakly h-regular along K</italic> if <list list-type="order"><list-item><p><italic>h</italic> is <italic>t</italic>-regular along <italic>K</italic> and</p></list-item><list-item><p><italic>h</italic>|<italic>h</italic><sup>−1</sup>(<italic>K</italic>) : <italic>h</italic><sup>−1</sup>(<italic>K</italic>) → <italic>K</italic> is a simple homotopy equivalence.</p></list-item></list></p><p>Now we can state our problem.</p><p>Suppose that we are given a diagram of weak <italic>h</italic>-regularity problem (*) satisfying the following conditions: <disp-formula id="FD5-kmj-59-563"><math id="m8" display='block'><mrow><mtable><mtr><mtd><mrow><mo stretchy='false'>(</mo><mo>*</mo><mo stretchy='false'>)</mo></mrow></mtd><mtd><mrow><mi>f</mi><mo>:</mo><mo stretchy='false'>(</mo><msup><mrow><mi>W</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>,</mo><mo>∂</mo><mi>W</mi><mo stretchy='false'>)</mo><mo>→</mo></mrow></mtd><mtd><mo stretchy='false'>(</mo><mrow><msup><mrow><mi>V</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>,</mo><mo>∂</mo><mi>V</mi><mo stretchy='false'>)</mo></mrow></mtd></mtr><mtr><mtd><mrow/></mtd><mtd><mrow/></mtd><mtd><mo>∪</mo></mtd></mtr><mtr><mtd><mrow/></mtd><mtd><mrow/></mtd><mtd><mo stretchy='false'>(</mo><mrow><msup><mrow><mi>M</mi></mrow><mi>m</mi></msup><mo>,</mo><mo>∂</mo><mi>M</mi><mo stretchy='false'>)</mo><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math></disp-formula><list list-type="simple"><list-item><p>(C.1) <italic>f</italic>|<italic>W</italic> : <italic>W</italic> → <italic>V</italic> is a (not necessarily simple) homotopy equivalence.</p></list-item><list-item><p>(C.2) <italic>f</italic> |<italic>∂W</italic> : <italic>∂W</italic> → <italic>∂V</italic> is weakly <italic>h</italic>-regular along <italic>∂M</italic>.</p></list-item><list-item><p>(C.3) <italic>f</italic>|<italic>∂W</italic> −<italic>f</italic><sup>−1</sup>(<italic>∂M</italic>) : <italic>∂W</italic> −<italic>f</italic><sup>−1</sup>(<italic>∂M</italic>) → <italic>∂V</italic> −<italic>∂M</italic> is a Λ′-homology equivalence, where <img src="http://pdf.medrang.co.kr/KMJ2/2019/059/kmj-59-563f3.jpg" class="inline_graphic" /> with <italic>π′</italic> = <italic>π</italic><sub>1</sub>(<italic>W</italic>). The Λ′<italic>-homology group</italic> of <italic>∂W</italic> −<italic>f</italic><sup>−1</sup>(<italic>∂M</italic>) means the integral homology of <italic>π</italic><sup>−1</sup>(<italic>∂W</italic> − <italic>f</italic><sup>−1</sup>(<italic>∂M</italic>)), where <italic>π</italic>: <italic>W̃</italic> → <italic>W</italic> is the universal covering.</p></list-item><list-item><p>(C.4) <italic>M</italic> is exterior 2-connected in <italic>V</italic>.</p></list-item></list></p></div><div><strong>Problem 2.4</strong><p>(The weak <italic>h</italic>-regularity problem (W.H.-R.P.)) <italic>When is f homotopic</italic> (<italic>rel. the boundary</italic>) <italic>to g</italic> : (<italic>W</italic>, <italic>∂W</italic>) → (<italic>V</italic>, <italic>∂V</italic>) <italic>which is weakly h-regular along M?</italic></p><p>Our purpose is to prove the following</p></div><div><strong>Theorem 2.5</strong><p><italic>There is a unique obstruction element γ</italic>(<italic>f</italic>) <italic>in P</italic><sub><italic>m</italic></sub>(ℰ) <italic>which vanishes if and only if f is homotopic</italic> (<italic>rel. ∂W</italic>) <italic>to such a map g</italic> (<italic>m</italic> ≧ 5). <italic>Here</italic> ℰ <italic>is the short exact sequence associated with</italic> (<italic>V</italic><sup><italic>m+2</italic></sup>, <italic>M<sup>m</sup></italic>).</p><p>For the algebraic definition of the group <italic>P</italic><sub><italic>m</italic></sub>(ℰ), see §5.1. Since <italic>P</italic><sub>2<italic>n</italic>+1</sub>(ℰ) ≅ <italic>L</italic><sub>2<italic>n</italic>+1</sub>(<italic>π′</italic>) ([<span class="xref"><a href="#B17">17</a></span>], the right-hand side being the Wall group [<span class="xref"><a href="#B32">32</a></span>]), we have</p></div><div><strong>Corollary 2.6</strong><p><italic>If m</italic> = <italic>odd</italic> ≧ 5, <italic>the obstruction is an element of the odd-dimensional Wall group L</italic><sub><italic>m</italic></sub>(<italic>π′</italic>).</p><p>This is independently obtained by Cappell and Shaneson [<span class="xref"><a href="#B4">4</a></span>]. The obstruction in this note and the one in our previous paper [<span class="xref"><a href="#B17">17</a></span>] are related as follows.</p></div><div><strong>Theorem 2.7</strong><p>(Restatement of Theorem 5.1 and Lemmas 5.3 and 5.4 of the present paper) <list list-type="order"><list-item><p><italic>Suppose we are given a diagram</italic> (*) <italic>of the weak h-regularity problem. Let T denote a tubular neighborhood of M</italic><sup>2<italic>n</italic></sup><italic>in V</italic><sup>2<italic>n</italic>+2</sup><italic>. Then f is homotopic</italic> (<italic>rel. ∂W</italic>) <italic>to a map g which is t-regular along</italic> ℱ<italic>T with g</italic><sup>−1</sup>(<italic>T</italic>) → <italic>T a simple homotopy equivalence.</italic></p></list-item><list-item><p><italic>Clearly g</italic><sup>−1</sup>(<italic>T</italic>) <italic>is a Poincaré thickening in the sense of [<span class="xref"><a href="#B17">17</a></span>]</italic>, <italic>and the obstruction η</italic>(<italic>g</italic><sup>−1</sup>(<italic>T</italic>)) <italic>to finding a locally flat spine is defined in P</italic><sub>2<italic>n</italic></sub>(ℰ)<italic>. The obstruction γ</italic>(<italic>f</italic>) <italic>to weak h-regularity is defined to be η</italic>(<italic>g</italic><sup>−1</sup>(<italic>T</italic>)).</p></list-item></list></p><p>In §6, we will prove the following “periodicity theorem” and the “invariance theorem under <italic>L</italic>-equivalence”.</p></div><div><strong>Theorem 2.8</strong><p>(Restatement of Theorem 6.1) <italic>Let id</italic><sub>ℂ</sub><sub><italic>P</italic><sub>2</sub></sub> × <italic>f be the diagram</italic><disp-formula id="FD6-kmj-59-563"><img src="http://pdf.medrang.co.kr/KMJ2/2019/059/kmj-59-563f4.jpg" class="inline_graphic" /></disp-formula><italic>then we have</italic><disp-formula id="FD7-kmj-59-563"><math id="m9" display='block'><mrow><mrow><mi>γ</mi></mrow><mo stretchy='false'>(</mo><msub><mrow><mrow><mtext>id</mtext></mrow></mrow><mrow><mi>ℂ</mi><msub><mrow><mi>P</mi></mrow><mn>2</mn></msub></mrow></msub><mo>×</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>ρ</mi><mo stretchy='false'>(</mo><mrow><mi>γ</mi></mrow><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>,</mo></mrow></math></disp-formula><italic>where ρ : P</italic><sub><italic>m</italic></sub>(ℰ) → <italic>P</italic><sub><italic>m</italic>+4</sub>(ℰ) <italic>is the algebraic periodicity isomorphism.</italic></p></div><div><strong>Theorem 2.9</strong><p>(Restatement of Theorem 6.2) <italic>If</italic><inline-formula><math id="m10" display="inline"><mrow><msubsup><mrow><mi>M</mi></mrow><mn>1</mn><mi>m</mi></msubsup></mrow></math></inline-formula><italic>and</italic><inline-formula><math id="m11" display="inline"><mrow><msubsup><mrow><mi>M</mi></mrow><mn>2</mn><mi>m</mi></msubsup></mrow></math></inline-formula><italic>are L-equivalent</italic> (<italic>rel. ∂</italic>), <italic>then γ</italic><sub><italic>M</italic><sub>1</sub></sub>(<italic>f</italic>) = <italic>γ</italic><sub><italic>M</italic><sub>2</sub></sub> (<italic>f</italic>). (<italic>The notation γ</italic><sub><italic>M</italic></sub> (<italic>f</italic>) <italic>is used to emphasize the submanifold M.</italic>)</p></div>
Surgery below the Middle Dimension
Dimension

Suppose the diagram (*) (on the previous page) is given. In this section we will perform surgery on f−1(M) below the middle dimension. The middle dimension will be studied in §4 and §5. In what follows, whenever we consider the preimage f−1(M), we are assuming that f is t-regular along M.

We will introduce the following notation:

Lm = f−1(Mm),

  • N; a regular neighborhood of Lm in Wm+2,

  • E; the exterior of N,

  • N; the frontier of N,

  • T; a regular neighborhood of Mm in Vm+2, (We assume N = f−1(T).)

  • F; the exterior of T,

  • T; the frontier of T, and Φ denotes the quadruple (EFNT).

  • Let α be an element of πi+2(Φ). Suppose ∂απi+1(E, ℱN) is represented by a normally embedded (i + 1)-handle H in W attached to L : (See [8]) H=Di+1×DmiWm+2,HLm=Di+1×Dmi.(∂α denotes the image of α under the homomorphism : πi+2(Φ) → πi+1 (E, ℱN).) Then we have

    Lemma 3.1

    There is a homotopy (rel. boundary) from f to f′which satisfies f′−1(M) = (LInt(∂Di+1 × Dmi)) ∪ Di+1 × ∂Dmi. In other words, the surgery can be performed by a homotopy of f.

    Although the proof is not difficult, it is long and tedious, so we omit it. An analogous argument is done in [15].

    To make Lm = f−1(Mm) Exterior 2-connected

    Suppose m ≧ 4. Since πi(F, ℱT)=0 (i ≦ 2) (this is our hypothesis (C.4)), we have :π2(Φ)π1(E,N),and:π3(Φ)π2(E,N)0.(These are obtained by the homotopy sequence of quadruple Φ.)

    Thus any element in πi(E, ℱN) (i ≦ 2) is of the form ∂α, where απi+1(Φ), and by dimension reasons, ∂α can be represented by a normally embedded i-handle. Then Lemma 3.1 applies.

    The effect of this surgery is to kill ∂α. Thus by successive surgeries, one can kill the whole sets πi(E, ℱN) (i ≦ 2). For a detailed proof, see [8].

    Hereafter we will assume that f−1(M) = L is exterior 2-connected and that the following diagram holds

    Surgery below the Middle Dimension

    Since f : Wm+2Vm+2 is a homotopy equivalence, the homomorphisms are surjective.

    The corresponding kernels are denoted by K* (N; Λ′), K* (E; Λ), , etc. (The meaning of the Λ-homology: For example, H* (E; Λ) denotes the integral homology of , the universal covering of E. Recall π1(E) ≅ π. Similarly H* (N; Λ′), the Λ′-homology group, is defined by the integral homology of Ñ, the universal covering of N, where π′ = π1(N).)

    Assume 3i+1[m2], and suppose that an element of is of the form ∂α, where απi+2(Φ), and is represented by a normally embedded handle H. Then Lemma 3.1 applies. f is deformed by a homotopy (rel. ) to f′ with f′1(M) a new submaifold which is obtained by the surgery along H. The corresponding new exterior (or frontier) is denoted by E′ (or ℱN′). Then we have,

    Lemma 3.2πj(E,N)=0(j2)Kj(E,N;Λ)Kj(E,N;Λ)(j<i+1)Ki+1(E,N;Λ)Ki+1(E,N;Λ)/Λ(α).

    If K* is replaced by H*, this is a standard result in codimension two surgery theory (cf. [8, 17]). The “standard proof” can apply to Lemma 3.2 under appropriate modifications.

    Denote the restrictions of f, f |E : EF, f|ℱN : ℱN → ℱT and f|L : LM by fE, f, fL respectively.

    The purpose of this paragraph is to prove the following:

    Theorem 3.3

    If m ≧ 4, we can perform ambient surgery on Lm = f−1(Mm) via homotopy (rel. ∂) of f to obtain a new map (again denoted by f) so that the new submanifold has the following properties

    Lm = f−1(Mm) remains exterior 2-connected,,

  • πi(Φ) = 0 (i[m2]+1),

  • πi(fE) = 0 (i[m2]+1),

  • πi(f) = 0 (i[m2])and

  • πi(fL) = 0 (i[m2]).

  • The next lemma is useful in proving Theorem 3.3.

    Lemma 3.4

    Let Lm = f−1(Mm) be exterior 2-connected. Let ℓ be an integer ≧ 1. Suppose

    πi(Φ) = 0 for i + 2,

  • πi(fE) = 0 for iℓ and

  • πi(fL) = 0 for iℓ.

    Then it follows that

  • πi(fE) = 0 for i + 2,

  • πi(f) = 0 for i + 1 and

  • πi(fL) = 0 for i + 1.

  • Proof

    From the homotopy exact sequence of Φ, we have πi(f)πi(fE)fori+1.

    This implies πi(f)=0fori,by(b).

    Let ϖ0 : ŴW (or ϖ1 :V) be the universal covering space of W (or V), and let Ê = ϖ0−1(E), N^=ϖ01(N), F^=ϖ11(F)and T^=ϖ11(T). Their fundamental groups are isomorphic to the cyclic group C with generator t.

    Let N^=ϖ01(N), L^=ϖ01(L), T^=ϖ11(T) and M^=ϖ11(M). These are connected and simply-connected.

    Let f^:N^T^, E : Ê, L : be the liftings of f, fE, fL, respectively.

    By (3.2) and hypothesis (b) together with the Hurewicz theorem, we have {H+1(f^E)π+1(f^E)/(1t)π+1(f^E),andH+1(f^)π+1(f^)/(1t)π+1(f^).

    Here (1 − t) is an element of , the integral group-ring of C, and we consider π+1(E) and π+1() as -modules.

    Now note that N^ (resp. T^) is the total space of an S1-bundle over (resp. ), so the homomorphism πi(N^)πi(L^) (resp. πi(T^)πi(M^)) induced by the projection is isomorphism for i ≧ 3, and injective for i = 2.

    From this and the homotopy exact sequences of f^:N^T^ and L : , we have {πi(f^)πi(f^L)isinjectivefori2.Ifi3,thisisanisomorphism,wherethemapisinducedbytheprojectionofS1bundles.The generator t of C is represented by a fiber of the S1-bundle, thus t acts trivially on πi(L). Therefore, by (3.4), t also acts trivially on πi(), in particular, on π+1(). The isomorphism (3.1) implies that the action of t is also trivial on π+1(E). This implies (1 − t)π+1() = 0 and (1 − t)π+1(E) = 0. Therefore we have from (3.3) and (3.1)Since the restrictions of f, fE, f, fN, fL are of degree 1, the homomorphisms , , and are all surjective. Let Ki(Ê), Ki(N^). Ki() and Ki() be the corresponding kernels.

    Consider the following diagram obtained by Mayer-Vietoris sequences:

    Here we have used the hypothesis that f : WV is a homotopy equivalence (C.1). By diagram chasing we see that Ki(N^)Ki(E^)Ki(N^)issurjective(i).

    On the other hand, from the homology exact sequences it follows that Ki(N^)Hi+1(f^), Ki(Ê) ≌ Hi+1(E) and Ki() ≌ Hi+1(N).

    This and (3.5) imply K(N^)K(E^)

    We need an elementary algebraic lemma.

    Lemma 3.5

    Let H, G1, G2be abelian groups, φ : HG1, ψ : HG2homomorphisms. Suppose that φ is injective and that φψ : HG1G2is surjective. Then G2 ≌ {0}.

    Proof

    Suppose there were a non-zero element xG2, and let yH be mapped under φψ to 0 ⊕ x. Then φ(y) = 0. Since φ is injective, y = 0 so x = ψ(y) = 0. This is a contradiction.

    By Lemma 3.5 together with (3.6) and (3.7), we have K(N^)=0.

    Note that 0 ≌ K() ≌ K() ≌ H+1(L). Our hypothesis (c): πi(fL) = 0 for iand the Hurewicz theorem imply π+1(L) ≌ H+1(L) ≌ 0. Thus πi(fL)0(i+1).

    This is the conclusion (f). From (3.9) and (3.4) follows πi(f)πi(f^)0fori+1.

    This is the conclusion (e). From (3.10) and (3.5) follows πi(fE)0fori+1.

    We see that K+1(N^)H+2(f^)π+2(f^)(Hurewicz.Recallthatthet-actiononπ+2(f^)istrivial.)π+2(f^L)(+23)H+2(f^L)(Hurewicz)K+1(L^)K+1(N^).

    Apply this to (3.6) and use Lemma 3.5. Then we have K+1(E^)=0.

    Consider the exact sequence of Φ: π+2(f^)π+2(f^E)π+2(Φ^)0by(a).From this we see that the t-action on π+2(E) is trivial, because its action on πℓ+2(f^E) is trivial. Thus 0 = K+1(Ê) ≌ H+2(E) ≌ π+2(E) ≌ π+2(fE). This together with (3.11) implies the conclusion (d). This completes the proof of Lemma 3.4.

    Proof of Theorem 3.3

    Cases where m = 4 or 5.

    First recall Namioka’s theorem [26]. We state it in our situation.

    Theorem 3.6

    (Namioka’s Theorem) LetΦ=(EfEFNfT). Suppose πi(F, ℱT) = 0 for i ≦ 2 and πi(fE) = 0 for ik (k ≧ 1).

  • πi(Φ) = 0 for 1 < ir if and only if Hi(Φ; Λ) = 0 for ir. Here 1 < rk + 2.

  • If 1 < rk + 1 and πi(Φ) = 0 for ir, the Hurewicz maph:πi(Φ)Hi(Φ;Λ)is isomorphic for ir + 1.

  • We now want to prove that our condition “Lm and Mm are exterior 2-connected” implies (ii) ∼ (v) of Theorem 3.3 for m=4,5([m2]=2).

    From Λ-homology exact sequence of Φ, we have Hi+1(Φ;Λ)Ki(E,N;Λ).Since πi(E, ℱ N) ≅ πi(F, ℱ T) = 0 i ≦ 2), Hi(E,N;Λ)Hi(F,T;Λ)0(i2)by the Hurewicz theorem.

    Thus Hi(Φ) ≅ Ki− 1(E, ℱ N; Λ) ≅ 0 for i ≦ 3. Since π1(fE) ≌ 0, we have πi(Φ) ≌ 0 for i ≦ 3 by Namioka’s theorem (I). (π1(fE) ≌ 0 follows from the diagram at the end of §3.1.) Then apply Lemma 3.4 with = 1 to obtain πi(fE) = 0 (i ≦ 3), πi(f) = 0 (i ≦ 2) and πi(fL) = 0 (i ≦ 2). This proves Theorem 3.3 for m = 4, 5.

    Cases where m ≧ 6.

    Inductively we assume πi(Φ) = 0 (i+ 2), πi(fE) = 0 (i+ 2), πi(f) = 0 (i+ 1) and πi(fL) = 0 (i+ 1). These assumptions in fact hold when = 1. Apply Namioka’s theorem (II) with k = + 1, r = + 2. Then the Hurewicz map h:π+3(Φ)H+3(Φ;Λ)is an isomorphism. Therefore any element of Kℓ+2(E, ℱ N; Λ) ≅ Hℓ+3 (Φ; Λ) can be represented by an element α of π+3(Φ). The “boundary” ∂α in π+2(E, ℱN) is proved to be represented by a normally embedded (+ 2)-handle H attached to Lm = f−1(M), provided that +2[m2] (See [8, Lemma 3.3]).

    Perform ambient codimension 2 handle exchange along H via homotopy of f (Lemma 3.1), then we have (by Lemma 3.2) Hi(Φ;Λ)Ki1(E,N;Λ)0(i+2),and H+3(Φ;Λ)K+2(E,N;Λ)K+2(E,N;Λ)/(α).Since H+3(Φ; Λ) is finitely generated over Λ, we will have H+3(Φ′; Λ) ≌ 0 after a finite number of the procedures above.

    Apply Namioka’s theorem (I) with k = + 2, r = + 3. Then we have πi(Φ)=0fori+3.Apply Lemma 3.4 with + 1 in place of , then we have πi(f′E) = 0 (i+ 3), πi(f′) = 0 (i+ 2) and πi(f′L) = 0 (i+ 2).

    Proceeding inductively we will have πi(Φ)=0(i[m2]+1),πi(fE)=0(i[m2]+1),πi(f)=0(i[m2]),andπi(fL)=0(i[m2]).

    This completes the proof of Theorem 3.3.

    Proof of Theorem 2.5 in the Odd Dimensional Case
    onal Case

    Suppose m = dim Lm = 2n + 1 ≧ 5; n=[m2]. Let Φ=(EfEFNfT).

    There are two exact sequences which contain πn+2(Φ); α)πn+2(Φ)πn+1(E,N)πn+1(F,T),β)πn+2(Φ)πn+1(f)πn+1(fE).

    In §3 we performed surgery below the middle dimension, thus we may assume πi(fL)=0forin,πi(fE)=0forin+1,πi(f)=0forin(Theorem3.3).

    Then from β), we have β)πn+2(Φ)πn+1(f)0.

    By (3.4) in the proof of Lemma 3.4, we have πn+1(f)πn+1(fL)Hn+1(fL;Λ)Kn(L;Λ). Thus β′) becomes β)πn+2(Φ)Kn(L;Λ)0.(isusedbyabuseofthenotation.)

    Our task is to make fL a simple homotopy equivalence, and fL is already n-connected. Thus we have only to kill Kn(L; Λ′). Wall’s surgery theory [32] tells us that there is a unique obstruction θ(fL) in Lm(π′) to killing the group Kn(L; Λ′) by abstract Wall surgery. θ(fL) is defined by the surgery obstruction of the following normal map diagram: (***)νL(f¯)*νW|MTLmfLMmwhere : VW is a homotopy inverse of f : WV and νL (resp. νW) the normal bundle of L (resp. W).

    We will show that θ(fL) is also the obstruction to make fL a simple homotopy equivalence by codimension 2 ambient surgery through a homotopy of f (rel. the boundary).

    First suppose θ(fL) = 0. Then following [32], we can find a finite number of disjoint embeddings gi : Sn × Dn+1Lm, each joined by a path to a base point, having the property that if we perform surgery on them, the resulting fL : LM will be a simple homotopy equivalence. All of them represent elements of Kn(L; Λ′). Denote the elements by [gi]. By β″), there are elements i in Φn+2(Φ) such that ∂′g̃i = [gi]. Let be the boundary homomorphism in α). Then ∂g̃i are elements of πn+1(E, ℱN) which are represented by a map hi : (Dn+1, Sn) → (E, ℱN). In [8, Lemma 3.3] we proved that in the odd dimensional case where m = 2n + 1 hi’s can be represented by normally embedded (n + 1)-handles Hi=Din+1×Din+1.

    Moreover, it is easily verified that the attaching framed n-spheres HiLm=Din+1×Din+1 can be taken to coincide with any framed n-spheres in Lm if they represent the same homotopy classes as HiLm’s. Thus we may assume HiLm = gi(Sn×Dn+1). Then Lemma 3.1 can be applied, and there is a homotopy (rel. ∂W) of f to a map f′ which realizes the surgery on the embedded spheres gi(Sn ×Dn+1) as the ambient codimension 2 surgery along the (n + 1)-handles Hi; f1(Mm)=(LmiIntgi(Sn×Dn+1))iDin+1×Din+1.The resulting fL′ : L′M is clearly a simple homotopy equivalence.

    Conversely, suppose that f is homotopic (rel. ∂W) to a map f′ such that f|f1(M):f1(M)Mis a simple homotopy equivalence. Then by the transverse regularity theorem, we can construct a normal cobordism between (***) and a normal map which is a simple homotopy equivalence. Thus θ′(fL) = 0. This completes the proof of Theorem 2.5 in the odd dimensional cases.

    The Even Dimensional Case
    onal Case

    Suppose m = dim Lm = 2n ≧ 4. The sequences α) β) β′) β″) remain valid in this case.

    In order to prove Theorem 2.5 in the even dimensional case, we first introduce the notion of geometric free cores (Cf. [17, Proof of Lemma 5.8]).

    Let as before. Wall [32, Lemma 2.3] proved that Kn(L; Λ′) is a stably free Λ′-module with a preferred equivalence class of s-basis. We may assume that Kn(L; Λ′) is actually Λ′-free, for after performing codimension 2 surgery on some trivial n-handles, we can change Kn(L; Λ′) into Kn(L; Λ′)⊕(standard planes). Here “A trivial n-handle” means an n-handle representing the zero element of πn(E, ℱN). See Wall [32, Lemma 5.5]. Let e1, e2, . . . , er be the preferred Λ′-basis of Kn(L; Λ′). Then they are lifted to elements 1, 2, . . . , r of πn+2(Φ). See sequence β″). By the standard technique, we can represent the elements ∂1, ∂2, . . . , ∂r of πn+1(E, ℱN) by “pathed” disjoint embeddings gi : (Dn+1, ∂Dn+1) → (E, ℱN), i = 1, . . . , r. (This technique is explained in [32, pp. 39–40].) Take a regular neighborhood Ri of gi(Dn+1) in E and construct a submanifold N(i=1rRi) in Wm+2. This submanifold is called a geometric free core of f : WV and is denoted by W*. A geometric free core has some useful properties.

    In order to state the properties we introduce some notations:

    U*=WW*¯, the complement of Int W*,

  • W* = U*W*, the frontier of W*

  • N*=12N, the tubular neighborhood of Lm in W* (we may assume that the radius of N* is a half of that of N),

  • E*=W*N*¯, the exterior of N* in W*,

  • N* = N*E*, the frontier of N*.

  • Theorem 5.1

    • f : WV is homotopic (rel. ∂W) to a map (again denoted by f) which is t-regular alongT 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.

    • πi(E*, ℱN*) = 0 for in, and πn+1(E*, ℱN*) is a free Λ-module with the basis ∂ẽ1, . . . , ∂ẽr.

    Remark 5.2

    Λ′-homology of F is defined to be in the notation of the proof of Lemma, similarly for Λ′-homologies of U*, ℱW* or ℱT.

    Proof of Theorem 5.1
    • 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 Ri’s (representing elements of πn+1(E, ℱN)) are mapped to zero in πn+1(F, ℱT) by f, then we can construct the desired homotopy which “splits” along T.

      The condition is satisfied in our situation; the core disks of Ri’s are of the form ∂ẽi, iπn+2(Φ), and so they are mapped to zero in πn+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 ψ : LW* be the inclusion, φ : LT the composition LfLM~T. Then we have the diagram we identify the mapping cylinder of φ with T.

    According to [32, Lemma 2.5], in order to prove that f|W* is a simple homotopy equivalence, we have only to show that θ is a simple homotopy equivalence. Let ℋ be the Λ′-homology sequence of C*(ψ)θC*(φ), then we have τ(C*(φ))=τ(C*(ψ))+τ(θ)+τ()([23, Theorem 3.2]), where τ denotes the Whitehead torsion. The only non-zero Λ′-homology of C*(ψ) is Hn+1(ψ) which is based isomorphic to Kn(L; Λ′). This follows from the construction of W*. On the other hand the only non-zero Λ′-homology of C*(φ) is Hn+1(φ) which is only non vanishing kernel Kn(L; Λ′) of fL : LM. θ* induces the identity of these two Kn(L; Λ′)’s by the construction of W*. Thus we have τ(ℋ) = 0.

    The bases of Hn(φ) and of Hn+1(ψ) are chosen so that τ(C*(φ)) = τ(C*(ψ)) = 0 ([32, p. 27]). Hence τ(θ) = 0 follows as desired.

    (iii) and (iv) follow easily from (ii) and the hypothesis that f : WV is a homotopy equivalence.

  • (v) is obvious by the construction of W*.

  • The proof of Theorem 5.1 is completed.

    Because of property (ii) of W* (Theorem 5.1), W* is an m-Poincaré thickening in the sense of [17, Definition 1.1]. Thus there is a unique obstruction η(W*) in Pm(ℰ) to finding a locally flat spine of W* [17]. We assert that η(W*) serves as the obstruction of the weak h-regularity problem as well.

    Lemma 5.3

    If η(W*) = 0, f is homotopic (rel. ∂W) to a map denoted by the same letter f with f|f−1(M) : f−1(M) → M a simple homotopy equivalence. (m ≧ 6).

    Lemma 5.4

    The element η(W*) ∈ Pm(ℰ) does not depend on a particular choice of a geometric free core W*and depends only on the diagramf:(W,W)(M,M).(V,V)

    These lemmas will be proved in §5.2.

    Seifert Forms

    Let us recall the definition of Seifert (−1)n-forms introduced in [17]. Let =[1Cπϖ*π1] be associated with the pair (W*, L); π = π1(ℱN*) ≌ π1(E*), and π′ = π1(L). The homomorphism ϖ* : π → π′ is induced by the projection ϖ of the S1-bundle ℱN*L. C is a cyclic group with a specified generator t. Let Λ be , and let Λ′ be , as usual. Abelian groups Qnt(π) and Qn(π′) are defined by Qnt(π)=Λ/{a(1)na¯t|aΛ},Qn(π)=Λ/{b(1)nb¯|bΛ},where ¯ : Λ → Λ (or Λ′ → Λ′) is induced by gg1 for gπ or (by gg−1 for gπ′). Here Qn(π′) is Wall’s notation [32]. The homomorphism ϖ* : ππ′ induces homomorphisms ϖ* : Λ → Λ′ and ϖ*:Qnt(π)Qn(π).

    A triple (G, λ, µ) consisting of a finitely generated free Λ-module G and maps λ : G × G → Λ, μ:GQnt(π) is called a free Seifert (−1)n-form over ℰ if it satisfies the following ([17]):

    λ(x,y)=(1)nλ(y,x)¯t

  • For any fixed y, λ(*, y) : G → Λ is a Λ-homomorphism.

  • µ(x + y) = µ(x) + µ(y) + λ(x, y).

  • λ(x,x)=μ(x)+(1)nμ(x)¯t.

  • µ(ax) = (x)ā for ∀a ∈ Λ.

  • Λ′⊗ΛG (denoted by G′) is a free Λ′-module with a preferred basis {ei}, and G′ has a structure of Wall’s special Hermitian (−1)n-form (λ0, µ0).

  • The following diagrams are commutative: where : GG′ is defined by (x) = 1 ⊗ x. Sometimes (G′, λ0, µ0) is denoted by Λ ⊗Λ (G, λ, µ).

  • A free Seifert (−1)n-form (G, λ, µ) is null-cobordant if there is a sub Λ-module H of G such that λ(H × H) = 0, µ(H) = 0 and H is mapped onto a subkernel of G′ under : GG′.

    (G, λ, µ) is stably null-cobordant if a direct sum of (G, λ, µ) and a finite number of “standard planes” defined by (Λx ⊕ Λy, λ, µ) with λ(x, y) = 1, λ(y, x) = (−1)nt, µ(x) = µ(y) = 0, is null-cobordant. (Note that our “standard plane” is not the same as in the Wall’s book [32].)

    Definition 5.5

    ([17]) The group P2n() is defined to be the Grothendieck group of all free Seifert (−1)n-forms over ℰ modulo the subgroup generated by all stably null-cobordant forms. The group structure is given by the direct sum ⊕.

    Remark 5.6

    P2n+1(ℰ) is defined to be L2n+1(π′).

    Remark 5.7

    Note that (G, λ, µ) represents the zero of P2n(ℰ) if and only if there is a stably null-cobordant form Y such that (G, λ, µ) ⊕ Y is stably null-cobordant. However we have proved, in [17, Lemma 5.3], that it is equivalent to saying that (G, λ, µ) itself is stably null-cobordant.

    Geometric Meaning of Seifert Forms

    Let W* be a free core of f : WV. We proved in [17] that the Λ-module πn+1(E*, ℱN*) carries a structure of a Seifert (−1)n-form (λ, µ). (See Appendix of the present paper.)

    The following properties are important to our purpose:

    Λ′⊗Λπn+1(E*, ℱN*) is identified with Kn(L; Λ′), and Λ′ ⊗Λ (λ, µ) with (λ0, µ0), Wall’s Hermitian form on Kn(L; Λ′).

  • Elements x1, x2, . . . , xs of πn+1(E*, ℱN*) can be represented by mutually disjoint normally embedded (n + 1)-handles if and only if µ(xi) = 0 (∀i), and λ(xi, xj) = 0 (∀i, j), (2n ≧ 6).

  • We can now prove Lemma 5.3.

    Proof of Lemma 5.3

    The obstruction η(W*) ∈ P2n(ℰ) is represented by the free Seifert (−1)n-form (πn+1(E*, ℱN*), λ, µ). Suppose η(W*) = 0, then by Remark 5.7 above, the Seifert form (πn+1(E*, ℱN*), λ, µ) is stably null-cobordant. After performing codimension 2 surgery along some trivial n-handles, we may suppose it is actually null-cobordant (See [17, Lemma 4.6].). Then there is a sub Λ-module H which is mapped onto a subkernel of Kn(L; Λ′), satisfying λ(H × H) = 0, µ(H) = 0. Let e1, . . . , er be the basis of the subkernel.

    These e1, . . . , er are lifted to some 1, …,r in Hπn+1(E*, ℱN*). By the geometric property (ii) of Seifert forms, 1, …,r can be represented by mutually disjoint normally embedded (n + 1)-handles H1, . . . , Hr. If F*, T* are defined by T*=12T, F*=TT*¯, it is clear that πi(F*, ℱT*) = 0 for all i. Thus the quadruple ple(E*F*N*T*) denoted by Φ*, satisfies πi+2(Φ*)πi+1(E*,N*) for all i. Lemma 3.1 can now be applied to H1, . . . , Hr, and f|W* : W*T ; we can find a homotopy (rel. ∂W*) of f |W* to a map f′ with f′1(M) the resulting submanifold obtained by performing surgery on the framed n-spheres LmHiSn × Dn+1 representing e1, . . . , er. Since {e1, . . . , er} is the preferred basis of a subkernel of Kn(L; Λ′), the map f′−1(M) → M must be a simple homotopy equivalence [32]. Extending the homotopy of f|W*to the whole of W by the identity, we will obtain the desired homotopy.

    Proof of Lemma 5.4

    Let W1* be another free core of f : WV. We have to prove that η(W*)=η(W1*)P2n(). There is a homotopy of f (rel. ∂W) “between W* and W1*”; this means that there is a map H : W × IV × I with f = H|W × {0} : W × {0} → V × {0}, f′ = H|W × {1} : W × {1} → V × {1}, and f−1(T × {0}) = W* and f1(T×{1})=W1* hold.

    As we proved in [8], the submanifold M × I in V × I can be transformed into an exterior n-connected submanifold Y2n+1 by exchanging handles in codimension 2 (rel. (M × I)). Let U be a tubular neighborhood of Y in V × I, B the exterior of U; B=V×IU¯. U can be taken so that U ∩ (T × {i}) = T* × {i} (i = 0, 1). Let ℱU be the frontier of U; ℱU = UB.

    Now consider the preimage Z2n+1 = H−1(Y)⊂ W × I. By exchanging handles in codimension 2, Z2n+1 can be made exterior n-connected, and this surgery can be carried out as surgery through a homotopy of H (rel. (W × I)). This is because Y2n+1 is already exterior n-connected, so any normally embedded i-handle in W × I attached to Z2n+1 shrinks if it is mapped into (B, ℱU), provided that in. Therefore Lemma 3.1 is applied.

    Let Q denote a tubular neighborhood of Z in W × I, ℱQ the frontier of Q, and let P be the exterior of Q.

    Let Ψ=(E*E1*PN*N1*Q), Ψ=(F*F1*BT*T1*U), where E1*,N1* (or F1*, T1*) denote the exterior and the frontier in W1* (or T1). By the exterior n-connectivity of Y and Z, we have Here we used the fact that πi(F*,T*)πi(F1*,T1*)=0(i). Note that the map H is of degree one, hence the vertical maps are onto. We can kill the kernel of Hn+1(Ψ; Λ) → Hn+1(Ψ′; Λ) and make it an isomorphism. Here we will give an indication of it. By Namioka’s theorem, any element of Hn+1(Ψ; Λ) is represented by a suitable embedded n+1-disk g:(Dn+1;D+n,Dn)(P;E*,Q), where D+n (or Dn) is the upper (or lower) hemisphere of ∂Dn+1, with f|D+n:(D+n,D+n)(E*,N*) being null-homotopic. By “attaching a collar” to g(Dn) we extend the embedding g to an embedding g¯:(Dn+1;D+n,Dn)(W×I;W*,Z) which satisfies g¯(Dn+1)Z=g¯(Dn). Then this extends to a“normally embedded knob” g˜:(Dn+1;D+n,Dn)×Dn+1(W×I;W*,Z) satisfying g˜(Dn+1×Dn+1)Z=g˜(Dn×Dn+1). (See the proof of [17, Proposition 4.11].) If the first embedding g:(Dn+1;D+n,Dn)(P;E*,Q) represents an element in the kernel of Hn+1(Ψ; Λ) → Hn+1(Ψ′; Λ), it is proved that we can construct a homotopy (rel. (W×I)W*×{0}¯ from H to a map H′ which satisfies H1(Y)=Zg˜(Dn×Dn+1)¯g˜(Dn+1×Dn+1).The construction of the homotopy is essentially the same as in the proof of Lemma which was omitted. Repeating the above “knob exchanging” process, we can kill the kernel of Hn+1(Ψ; Λ) → Hn+1(Ψ′; Λ).

    The effect of the surgery on Hn+1(E*, ℱN*) is to add a direct sum of standard planes, and this does not affect the class in P2n(ℰ) which it represents. (Cf. the proof of [17, Proposition 4.11].) And we remain using the same notations E*, ℱ*N.

    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 Λ-coefficients): Hn+1(E*;N*)Hn+1(E1*;N1*)Hn+1(P,Q)Hn+1(B,U)0.

    Note that the kernel of Hn+1(P, ℱQ) → Hn+1(B, ℱU) is Kn+1(P, ℱQ) by the definition, so we have (**)Hn+1(E*,N*)Hn+1(E1*,N1*)Kn+1(P,Q)0.Let K denote the kernel of this surjection. We have Hn+1(E*,N*)Hn+1(E1*,N1*)πn+1(E*,N*)πn+1(E1*,N1*)by the Hurewicz theorem, and it has the Seifert (−1)n-form representing η(W)*η(W1)*. It is shown in [17, Theorem 3.5, Proposition 4.11] that the Seifert form vanishes on the sub Λ-module K.

    Notice that Λ′⊗ΛHn+1(E*, ℱN*) ≌ Kn(L, Λ′), ΛΛHn+1(E1*,N1*)Kn(L1,Λ). Tensoring Λ′ with (**), we have ΛΛKKn(L;Λ)Kn(L1;Λ)Kn(Z,Λ)0.This shows (by Wall [32, Proof of 5.7]) that Λ′⊗ΛK is mapped onto the subkernel of Kn(L; Λ′) ⊕ Kn(L1; Λ′).

    Thus the Seifert form on πn+1(E*,N*)πn+1(E1*,N1*) is null-cobordant by definition, and the element η(W*)η(W1*) which it represents is zero in P2n(ℰ), i.e., η(W*)=η(W1*). This completes the proof of Lemma 5.4.

    Now Theorem 2.5 follows immediately from Lemmas 5.3 and 5.4; we define γ(f) in the theorem to be η(W*) in the even dimensional case. The proof of Theorem 2.5 is completed.

    Some Properties of the Obstruction
    structionGeometric Periodicity
    Theorem 6.1

    Let(f:(Wm+2,W)(Vm+2,V)(Mm,M))be a diagram of the weak h-regularity problem with m ≧ 4 satisfying (C.1) ∼ (C.4) in §2. Letbe the associated extension. Then we haveγ(idP2×f)=ρ(γ(f))Pm+4(),where(idP2×f:P2×(Wm+2,W)P2×(Vm+2,V)P2×(Mm,M))is the product with the complex projective planeP2, and ρ : Pm(ℰ) → Pm+4(ℰ) is the algebraic periodicity isomorphism.

    Proof

    In the odd dimensional case, this is obvious by the definition of γ(f) = θ(fL). In the even dimensional case, we have defined γ(f) by η(W*) with W* a geometric free core of f. The product ℂP2 × W* is clearly a geometric free core of idP2 × f. Therefore, we have γ(idP2×f)=η(idP2×W*)=ρ(η(W*))[17,Theorem5.12]=ρ(f).

    The Invariance of γ(f) under L-equivalence of M

    Let f be as in Theorem 6.1. We will say that f splits along M if f is homotopic (rel. ∂W) to a map f′ which is weakly h-regular along M. The obstruction γ(f) in Theorem 2.5 is written here as γM(f) to emphasize the submanifold M.

    Theorem 6.2

    Let(Mim,Mi)(Vm+2,V) (i = 1, 2) be exterior 2-connected submanifolds of (V, ∂V) satisfying conditions (C.1) ∼ (C.4) (of §1). Suppose that ∂M1 = ∂M2and that M1and M2are L-equivalent (rel. the boundary) to each other in the sense of Thom. Then we haveγM1(f)=γM2(f)Pm(),(m ≧ 4).

    Corollary 6.3

    If m ≧ 5, f splits along M1if and only if it splits along M2.

    Outline of the proof of Theorem 6.2

    Since M1 and M2 are L-equivalent, there exists a submanifold Y2n+1 in V × I such that YV×{0}=M1×{0},YV×{1}=M2×{1},andYV×I=M1×I.Then the rest of the proof is completely the same as Lemma 5.4 or, in the odd dimensional case, as in §4.

    We will collect some known results on the structure of the group Pm().

    Theorem 6.4

    • P2k+1(1 → Cππ′ → 1) = L2k+1(π′)

    • Pm(11πidπ1)=Lm(π), in particular Pm(1 → 1 → 1 → 1 → 1) = Lm(1),

    • (Cm−1is the knot cobordism group of (m − 1, m + 1)-knots.)

    The isomorphism (i) is the definition of P2k+1(ℰ). The isomorphisms (ii) and (iii) are proved in [17].

    Let Pm(C) denote the group Pm(1 → CC → 1 → 1) for simplicity.

    Theorem 6.5

    P4k+2(C) is infinitely generated if C is a cyclic group of even (or infinite) order.

    For the proof of Theorem 6.5, see [17].

    Let ℰ denote an extension {1Cjππ1} with j the inclusion. Let ρ be the morphism defined by the following diagram:

    Theorem 6.6

    If π′is a finite group, then all elements in the kernel of ρ* : Pm(C) → Pm(ℰ) are of finite order whose orders divide the order of π′. Here ρ*is the homomorphism induced by ρ.

    Corollary 6.7

    Ifand the order of π′is odd, then the homomorphismis injective.

    Proof of Corollary 6.7

    According to Theorem 6.4 (iii), . Levine [11, 12] has proved that Cm −1 contains no odd torsions. Thus Corollary 6.7 follows from Theorem 6.6.

    Proof of Theorem 6.6

    By the periodicity of Pm(ℰ), we may assume that m is sufficiently large. Let ξ : EMm−1 be a closed 2-disk bundle over an (m − 1)-manifold which “represents” ℰ, i.e., the short exact sequence derived from the homotopy exact sequence of the associated circle bundle ∂EM is isomorphic to ℰ. In the following, M is identified with the zero section of EM.

    It is proved in [17] that is onto and any element σ of Pm(C) is represented by a PL (m − 1, m + 1)-knot κ = (Σm−1, Sm+1). Taking a cone over κ = (Σm−1, Sm−1), we have a (not necessarily locally flat) disk pair (Δm, Dm+2). Consider a pairwise boundary connected sum of (E × I, Mm−1×I) and (Dm+2, Δm). Then it is clear that (W, ((M × I)Δm)) = ((E × I)♮Dm+2, ((Mm−1 × I)Δm)) is an m-Poincaré pair.

    Here we recall the main result of [17]: An oriented manifold pair (Wm+2, Km−1) with Km−1∂Wm+2 is called an m-Poincaré thickening pair if K is locally flat in ∂W and the pair is simple homotopy equivalent to an m-Poincaré pair. A locally flat submanifold Lm of Wm+2 is said to be a (locally flat) spine if ∂Lm = Km−1 and the inclusion LmWm+2 is a simple homotopy equivalence. The main result of [17] states the following: Given an m-Poincaré thickening pair (Wm+2, Km−1) with m ≧ 5, there exists a well-defined obstruction element η in Pm() which vanishes if and only if the pair (Wm+2, Km−1) admits a spine (cf. Appendix below).

    Now returning to our present situation, we have η = ρ*(σ) with ρ* : Pm(C) → Pm(ℰ), where the element σPm(C) is represented by the PL (m − 1, m + 1)-knot κ = (Σm−1, Sm+1) taken above. (This follows from the naturality of our obstruction [17, § 5, Complement 1] and the construction of W = (E × I)♮Dm.)

    Suppose that the element σPm(C) is in the kernel of ρ* : Pm(C) → Pm(ℰ). Then η = ρ*(σ) = 0, thus by the main result of [17], a spine Lm of (W, ((M × I)♮Δm)) can be found.

    Let π : W be the universal covering of W. It is easy to see that the associated extension ℰ̃ of the m-Poincaré thickening (, π−1(∂Lm)) is 1 → CC → 1 → 1. Clearly W˜=E×l˜#(Dm+2,Δm) where = |π′| is the order of π′ = π1(W) ≌ π1(M). Thus the obstruction to finding a spine of (, π−1(∂Lm)) is · σPm(C). However, we have already such a spine = π−1(Lm). Therefore, ·σ = 0. This completes the proof of Theorem 6.6.

    Appendix

    In [17] we show that a kind of intersection form can be defined on πn+1(E, ℱN), which is associated with an even-dimensional submanifold of codimension two. The form is called the Seifert form, and it plays an essential role in the present paper. In this appendix, we will recall the geometric definition of it.

    Let L2n be a 2n-dimensional (locally flat) submanifold of a (2n+2)-dimensional manifold W2n+2. We suppose that it is exterior 2-connected. Thus a short exact sequence ℰ is associated with the manifold pair: ={1Cππ1},where π = π1(WL), π′ = π1(W), C = Coker(π2(W) → π1(S1)). (Cf. §2.) Let E be the exterior of a regular neighborhood N of L, ℱN the frontier of N : ℱN = NE. Then we have π = π1(E) ≌ π1(L). A map f : (Dn+1, Sn) → (E2n+2, ℱN) is said to be a nice immersion if

    f is a generic immersion in the sense of Haefliger. Thus f |IntDn+1 has only a finite number of isolated double points at which f(Dn+1) intersects with itself transversely,

  • f |Sn : Sn → ℱN is an embedding, and

  • the composition ϖ ○ (f|Sn) : SnL2n is a generic immersion, where ϖ is the projection map of the S1-bundle ℱNL2n.

  • A nice immersion f is pathed if a path γ(f) in ℱN from a base point * ∈ ℱN to a point in the image f(Sn) is specified.

    Two nice immersions f, g : (Dn+1, Sn) → (E2n+2, ℱN) intersect nicely if

    f(Dn+1) and g(Dn+1) intersect in general position,

  • f(Sn) ∩ g(Sn) = ∅ and

  • ϖf(Sn) and ϖg(Sn) intersect in general position.

  • Assume that two pathed nice immersions f and g intersect nicely. Then we will define a pairing λ(f, g) as an element of as follows:

    Let {p1, · · · , pk} be the set of intersection points of ϖf(Sn) and ϖg(Sn) in L2n. Let εi(f, g) be the sign ±1 of the intersection at pi. We are assuming that the S1-fiber of ϖ is oriented. The orientation convention will be described later. We take a following loop i(f, g) in ℱN : i(f,g)={*γ(f)pif(alongtheS1fiberϖ1(pi)inthepositivedirection)pigγ(g)1*},where pif (or pig) is the point of f(Sn) (or g(Sn)) over pi, i.e., {pif}=f(Sn)ϖ1(pi)N (or {pig}=g(Sn)ϖ1(pi)N). Let gi(f, g) ∈ π1(ℱN) be represented by the loop i(f, g).

    An auxiliary pairing is defined by α(f,g)=i=1kεi(f,g)gi(f,g).

    In order to define the pairing λ(f, g) we need another auxiliary pairing β(f, g), which is defined as follows: Let {q1, . . . , q} be the set of intersection points of f(Dn+1) and g(Dn+1) in E2n+2, ε′i the sign ±1 of the intersection at qi.

    Let g′i(f, g) ∈ π1(E) (≌ π1(ℱN)) be defined by the following loop in E: gi(f,g)={*γ(f)qiγ(g)1*}.]Then the pairing β(f, g) is defined by β(f,g)=i=1εi(f,g)gi(f,g).Now the pairing λ(f, g) is defined by the following formula: λ(f,g)=α(f,g)+(1)n+1(1t)β(f,g).Here t denotes the positive generator of the cyclic group C,

    Next we will define an element μ(f)Qnt(π) which corresponds to the self-intersection of f.

    Let α(f), be defined analogously to the definition of α(f, g), β(f, g). To define α(f) and β(f) we have only to replace the “intersection points” in the definition of α(f, g) and β(f, g) by the “self-intersection points” with an order of the two branches of ϖf(Sn) (or f(Dn+1)) arbitrarily fixed at each self-intersection point. If the order is reversed, α(f) and β(f) change. However it is proved in [17] that the ambiguity of α(f) and (1 − t)β(f) is contained in the subgroup I = {a − (−1)nā ·t|a ∈ Λ} of , so α(f) and (1 − t)β(f) are well-defined as elements of Qnt(π)=Λ/I. An integer is defined as follows: Let v be a non-singular vector field over ℱN which is along S1-fibers in their positive directions.

    By the condition (iii) of nice immersions, v is transverse to f(Sn). Let be the obstruction to extending this non-zero cross-section of the normal bundle of f(Sn) in ℱN to a non-zero cross-section of the normal bundle of f(Dn+1) in E. At this point we use the orientation conventions which will be stated in § A.1 below. Finally, μ(f)Qnt(π) is defined by the following: μ(f)=α(f)+(1)n+1(1t)β(f)+(1)n+1O(f),where is considered to be an element of Qnt(π) via .

    A.1. Orientation Conventions

    For an oriented manifold X, we will denote by [X] its orientation, and by [X]p the local orientation at p. We are given [L2n] and [W2n+2]. Then [E] is defined by [E]p = [W]p (∀pE). [ℱN] is defined by , where is the inward normal direction of E at p ∈ ℱN. The positive direction [S1] of an S1-fiber is given by [ℱN] = [L2n] × [S1]. The normal fiber ℝn+1 of f(Sn) in ℱN is oriented by [ℱN[ = [f(Sn)] × [ℝn+1].

    Proposition A.1

    ([17, Theorems 2.5, 2.9]) Under the above orientation conventions, λ(f, g) and µ(f) depend only on the homotopy classes of f and g.

    Thus we can define the following maps. λ:πn+1(E,N)×πn+1(E,N)Λ,μ:πn+1(E,N)Qnt(π).This completes the geometric definition of Seifert forms.

    Remark A.2

    πn+1(E, ℱN) is not necessarily a free Λ-module. Moreover, even if it is Λ-free, the bilinear form λ is not in general unimodular. The deviation from unimodularity “measures” the extent to which the submanifold L2n is knotted. The “determinant” det λ is somewhat like the Alexander polynomial. The property (a) of Seifert forms (§5.1) corresponds to the reciprocity of Alexander polynomials, and property (f) corresponds to the fact Δ(1) = ±1 in the classical theory.

    Acknowledgments and Postscript

    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 Topology. But it was rejected. I was very much disappointed, and lost the will to publish this note. Thirty years later, the note gave a motivation to define the “pull back relation”for non-spherical knots, which was investigated in our joint paper [2]. There we cited the note [16] under the title Note on the splitting problem in codimension two, because the word the weak h-regularity problem contained in the original title had already become difficult to understand its meaning.

    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 old-fashioned 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.

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