The Codimension Two Placement Problem and Homology Equivalent Manifolds

Cappell, Sylvain E.; Shaneson, Julius L.

doi:10.2307/1970901

Cited by 140 publications

(168 citation statements)

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“…Also, v p n (2a 4 n + 4a 2 n + 1) = 0 since 2a 2 n + 1 and 2a 4 n + 4a 2 n + 1 are relatively prime. So by ( * ) we have (2a 4 n + 4a 2 n + 1, −1) p n = 1. Suppose j < i.…”

Section: Realization Of Independent Discriminantsmentioning

confidence: 97%

Link concordance, homology cobordism, and Hirzebruch-type defects from iterated $p$-covers

Choon¹

2010

J. Eur. Math. Soc.

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Abstract. We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary depth of the derived series of the fundamental group, and can detect torsion which is invisible via signature invariants. Applications illustrating these features include the following: (1) There are infinitely many homology equivalent rational 3-spheres which are indistinguishable via multisignatures, η-invariants, and L 2 -signatures but have distinct homology cobordism types. (2) There is an infinite family of 2-torsion (amphichiral) knots with non-slice iterated Bing doubles; as a special case, we give the first proof of the conjecture that the Bing double of the figure eight knot is not slice. (3) There exist infinitely many torsion elements at any depth of the Cochran-Orr-Teichner filtration of link concordance.

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“…Also, v p n (2a 4 n + 4a 2 n + 1) = 0 since 2a 2 n + 1 and 2a 4 n + 4a 2 n + 1 are relatively prime. So by ( * ) we have (2a 4 n + 4a 2 n + 1, −1) p n = 1. Suppose j < i.…”

Section: Realization Of Independent Discriminantsmentioning

confidence: 97%

Link concordance, homology cobordism, and Hirzebruch-type defects from iterated $p$-covers

Choon¹

2010

J. Eur. Math. Soc.

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“…By obstruction theory, we see that ph ~ p since the compositions with t are homotopic. The same argument as in [CS,4.6] then shows that h is homotopic to a bundle map.…”

Section: By Lemma 18 A(h C) -0 and The Results Followsmentioning

confidence: 56%

“…locally flat, topologically locally flat) embedding of (V, N, N) into an Ai-cobordism (W, E, E) restricting to t and (|. Cobordism defines an equivalence relation; the cobordism classes of smooth (resp., P.L., topological) local knots in £ over / will be denoted by C0(f, £) (resp., CPL(/, U CTOP(/, £))• It can be shown that CH(idM, £) = CH(M, {) (as in [CS,§4]) if Wh(irxM) = {e}. In what follows, the assertions about "local knots" are meant to cover all three categories; the proofs of such assertions will usually apply as written in the smooth and PL categories and with modifications using [LR, KS] Proof.…”

Section: By Lemma 18 A(h C) -0 and The Results Followsmentioning

confidence: 99%

“…A triple a = ( H, <j>, ¡x ) will be called an r/-Hermitian form over 3F: Zir -> A if it is an rj-Hermitian form over Zir (see (Q1)-(Q5) of [CS,p. 286] is an isomorphism for ¡ < k -1 and surjective for i = k -1.…”

Section: Highly Connected Embeddings In Codimension Two By Susan Szczmentioning

confidence: 99%

“…Let tj = (-1)*. As in [W,§1;CS,§1], Hk+i(f;Zir) = Kk(M: Zir); intersection and self-intersection forms, <¡>: Kk(M) X Kk(M) -» Zir and p: Kk(M) -» 7(T|), respectively, are defined. As for the case of Zir coefficients, Hk+l(f; A) = Kk(M; A) also, and by [CS,1.4], Hk+l(f; A) = Hk+l(f; Zir) 0 A. Since/is it-connected and/|8W is A-homology (A: -l)-connected, it follows by arguments similar to those in [CS,§1] Proof.…”

Section: Highly Connected Embeddings In Codimension Two By Susan Szczmentioning

confidence: 99%

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Highly Connected Embeddings in Codimension Two

Szczepanski¹

1983

Transactions of the American Mathematical Society

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Abstract. In this paper we study semilocal knots over/into £, that is, embeddings of a manifold N into £(£). the total space of a 2-disk bundle over a manifold M, such that the restriction of the bundle projection^: E(£) -M to the submanifold N is homotopic to a normal map of degree one, f: N -N.We develop a new homology surgery theory which does not require homology equivalences on boundaries and, in terms of these obstruction groups, we obtain a classification (up to cobordism) of semilocal knots over/ into £.In the simply connected case, the following geometric consequence follows from our classification. Every semilocal knot of a simply connected manifold M#K in a bundle over M is cobordant to the connected sum of the zero section of this bundle with a semilocal knot of the highly connected manifold K into the trivial bundle over a sphere.Introduction. It is a well-known fact that complex hypersurfaces with isolated critical points give rise to embeddings of highly connected manifolds in spheres, that is, (real) codimension two submanifolds K of S2k+l with trt(K) -0 for i < k -1 [M]. To study such embeddings, it is natural to consider the techniques employed in the classification of classical knot theory and the higher-dimensional analoguesembeddings of S" into Sn+2. One approach is that of surgery theory. As one application of T-homology surgery theory, Cappell and Shaneson obtained a calculation of the knot cobordism groups [CS]. The more general problem classifying highly connected codimension two embeddings up to cobordism cannot be answered in terms of known surgery theories. One intent of this paper is the development and application of a new surgery theory, B-surgery, which is useful when studying geometric problems in which the dominant maps are not homotopy or homology equivalences. The maps derived from these situations will have the property that the restriction to the boundary does not induce a homology equivalence, not even over

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L²‐signatures, homology localization, and amenable groups

Choon

Orr

2012

Comm Pure Appl Math

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Aimed at geometric applications, we prove the homology cobordism invariance of the L2‐Betti numbers and L2‐signature defects associated to the class of amenable groups lying in Strebel's class D(R), which includes some interesting infinite/finite non‐torsion‐free groups. This result includes the only prior known condition, that Γ is a poly‐torsion‐free abelian group (or a finite p‐group). We define a new commutator series that refines Harvey's torsion‐free derived series of groups, using the localizations of groups and rings of Bousfield, Vogel, and Cohn. The series, called the local derived series, has versions for homology with arbitrary coefficients and satisfies functoriality and an injectivity theorem. We combine these two new tools to give some applications to distinct homology cobordism types within the same simple homotopy type in higher dimensions, to concordance of knots in three manifolds, and to spherical space forms in dimension 3. © 2012 Wiley Periodicals, Inc.

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The Codimension Two Placement Problem and Homology Equivalent Manifolds

Cited by 140 publications

References 21 publications

Link concordance, homology cobordism, and Hirzebruch-type defects from iterated $p$-covers

Link concordance, homology cobordism, and Hirzebruch-type defects from iterated $p$-covers

Highly Connected Embeddings in Codimension Two

L²‐signatures, homology localization, and amenable groups

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The Codimension Two Placement Problem and Homology Equivalent Manifolds

Cited by 140 publications

References 21 publications

Link concordance, homology cobordism, and Hirzebruch-type defects from iterated $p$-covers

Link concordance, homology cobordism, and Hirzebruch-type defects from iterated $p$-covers

Highly Connected Embeddings in Codimension Two

L2‐signatures, homology localization, and amenable groups

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L²‐signatures, homology localization, and amenable groups