Surgery obstructions on closed manifolds and the inertia subgroup

Hambleton, Ian

doi:10.1515/form.2011.090

Forum Mathematicum

2012

DOI: 10.1515/form.2011.090

|View full text |Cite

Surgery obstructions on closed manifolds and the inertia subgroup

Ian Hambleton

Abstract: Abstract. The Wall surgery obstruction groups have two interesting geometrically defined subgroups, consisting of the surgery obstructions between closed manifolds, and the inertial elements. We show that the inertia group I n+1 (π, w) and the closed manifold subgroup C n+1 (π, w) are equal in dimensions n + 1 ≥ 6, for any finitely-presented group π and any orientation character w : π → Z/2. This answers a question from [10, p. 107].

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2013

2014

Publication Types

Select...

Article2

Relationship

Self Cite0

Independent2

Authors

Journals

Cited by 2 publications

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Controlled homotopy equivalences and structure sets of manifolds

Hegenbarth

Repovš

2014

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

For a closed topological $n$--manifold $K$ and a map $p:K\to B$ inducing an isomorphism $\pi_1(K)\to\pi_1(B)$, there is a canonicaly defined morphism $b:H_{n+1}(B,K,\LL)\to \SS (K)$, where $\LL$ is the periodic simply-connected surgery spectrum and $\SS (K)$ is the topological structure set. We construct a refinement $a:H_{n+1}^{+}(B,K,\LL )\to \SS_{\varepsilon ,\delta }(K)$ 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, $H_{n+1}^{+}(B,K,\LL )\subset H_{n+1}(B,K,\LL )$, and $\SS_{\varepsilon ,\delta }(K)$ is the controlled structure set. We show that the Pedersen-Quinn-Ranicki controlled surgery sequence is equivalent to the exact $\LL$-homology sequence of the map $p:K \to B$, i.e. that $$H_{n+1}(B,\LL)\to H_{n+1}^{+}(B,K,\LL )\to H_n(K,\LL^{+})\to H_n(B,\LL ), \ \LL^{+}\to \LL,$$ is the connected covering spectrum of $\LL$. By taking for $B$ various stages of the Postnikov tower of $K$, one obtains an interesting filtration of the controlled structure set

show abstract

Controlled homotopy equivalences and structure sets of manifolds

Hegenbarth

Repovš

2014

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Surgery in codimension 3 and the Browder--Livesay invariants

Hegenbarth¹,

Muranov²,

Repovš³

2013

Turkish Journal of Mathematics

View full text Add to dashboard Cite

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

hi@scite.ai

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Surgery obstructions on closed manifolds and the inertia subgroup

Cited by 2 publications

References 29 publications

Controlled homotopy equivalences and structure sets of manifolds

Controlled homotopy equivalences and structure sets of manifolds

Surgery in codimension 3 and the Browder--Livesay invariants

Contact Info

Product

Resources

About