Journal of Topology最新文献

{"title":"A realisation result for moduli spaces of group actions on the line","authors":"Joaquín Brum,&nbsp;Nicolás Matte Bon,&nbsp;Cristóbal Rivas,&nbsp;Michele Triestino","doi":"10.1112/topo.12357","DOIUrl":"https://doi.org/10.1112/topo.12357","url":null,"abstract":"<p>Given a finitely generated group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>, the possible actions of <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> on the real line (without global fixed points), considered up to semi-conjugacy, can be encoded by the space of orbits of a flow on a compact space <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Y</mi>\u0000 <mo>,</mo>\u0000 <mi>Φ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(Y, Phi)$</annotation>\u0000 </semantics></math> naturally associated with <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> and uniquely defined up to flow equivalence, that we call the <i>Deroin space</i> of <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>. We show a realisation result: every expansive flow <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Y</mi>\u0000 <mo>,</mo>\u0000 <mi>Φ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(Y, Phi)$</annotation>\u0000 </semantics></math> on a compact metrisable space of topological dimension 1, satisfying some mild additional assumptions, arises as the Deroin space of a finitely generated group. This is proven by identifying the Deroin space of an explicit family of groups acting on suspension flows of subshifts, which is a variant of a construction introduced by the second and fourth authors. This result provides a source of examples of finitely generated groups satisfying various new phenomena for actions on the line, related to their rigidity/flexibility properties and to the structure of (path-)connected components of the space of actions.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142435192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Morse numbers of complex polynomials","authors":"Laurenţiu Maxim,&nbsp;Mihai Tibăr","doi":"10.1112/topo.12362","DOIUrl":"https://doi.org/10.1112/topo.12362","url":null,"abstract":"<p>To a complex polynomial function <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> with arbitrary singularities, we associate the number of Morse points in a general linear Morsification <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>f</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 <mo>:</mo>\u0000 <mo>=</mo>\u0000 <mi>f</mi>\u0000 <mo>−</mo>\u0000 <mi>t</mi>\u0000 <mi>ℓ</mi>\u0000 </mrow>\u0000 <annotation>$f_{t}:= f - tell$</annotation>\u0000 </semantics></math>. We produce computable algebraic formulae in terms of invariants of <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> for the numbers of stratwise Morse trajectories that abut, as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mo>→</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$trightarrow 0$</annotation>\u0000 </semantics></math>, to some point of the space or at infinity.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12362","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142435191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stated SL(\u0000 \u0000 n\u0000 $n$\u0000 )-skein modules and algebras","authors":"Thang T. Q. Lê,&nbsp;Adam S. Sikora","doi":"10.1112/topo.12350","DOIUrl":"https://doi.org/10.1112/topo.12350","url":null,"abstract":"&lt;p&gt;We develop a theory of stated SL(&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;annotation&gt;$n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;)-skein modules, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {S}_n(M,mathcal {N})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, of 3-manifolds &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; marked with intervals &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {N}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; in their boundaries. These skein modules, generalizing stated SL(2)-modules of the first author, stated SL(3)-modules of Higgins', and SU(n)-skein modules of the second author, consist of linear combinations of framed, oriented graphs, called &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;annotation&gt;$n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-webs, with ends in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {N}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, considered up to skein relations of the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;U&lt;/mi&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;l&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$U_q(sl_n)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-Reshetikhin–Turaev functor on tangles, involving coupons representing the anti-symmetrizer and its dual. We prove the Splitting Theorem asserting that cutting of a marked 3-manifold &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; along a disk resulting in a 3-manifold &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$M^{prime }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; yields a homomorphism &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 ","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142041631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups","authors":"Jason Behrstock,&nbsp;Mark Hagen,&nbsp;Alexandre Martin,&nbsp;Alessandro Sisto","doi":"10.1112/topo.12351","DOIUrl":"https://doi.org/10.1112/topo.12351","url":null,"abstract":"<p>We give a simple combinatorial criterion, in terms of an action on a hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We apply this to show that quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic (and even relatively hyperbolic in the genus 2 case). In genus at least three, there are no known infinite hyperbolic quotients of mapping class groups. However, using the hierarchically hyperbolic quotients we construct, we show, under a residual finiteness assumption, that mapping class groups have many nonelementary hyperbolic quotients. Using these quotients, we relate questions of Reid and Bridson–Reid–Wilton about finite quotients of mapping class groups to residual finiteness of specific hyperbolic groups.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Regularity of limit sets of Anosov representations","authors":"Tengren Zhang,&nbsp;Andrew Zimmer","doi":"10.1112/topo.12355","DOIUrl":"https://doi.org/10.1112/topo.12355","url":null,"abstract":"<p>In this paper, we establish necessary and sufficient conditions for the limit set of a projective Anosov representation to be a <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mi>α</mi>\u0000 </msup>\u0000 <annotation>$C^{alpha }$</annotation>\u0000 </semantics></math>-submanifold of the real projective space for some <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>∈</mo>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mn>2</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$alpha in (1,2)$</annotation>\u0000 </semantics></math>. We also calculate the optimal value of <span></span><math>\u0000 <semantics>\u0000 <mi>α</mi>\u0000 <annotation>$alpha$</annotation>\u0000 </semantics></math> in terms of the eigenvalue data of the Anosov representation.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Coarse cubical rigidity","authors":"Elia Fioravanti,&nbsp;Ivan Levcovitz,&nbsp;Michah Sageev","doi":"10.1112/topo.12353","DOIUrl":"https://doi.org/10.1112/topo.12353","url":null,"abstract":"<p>We show that for many right-angled Artin and Coxeter groups, all cocompact cubulations coarsely look the same: They induce the same coarse median structure on the group. These are the first examples of non-hyperbolic groups with this property. For all graph products of finite groups and for Coxeter groups with no irreducible affine parabolic subgroups of rank <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$geqslant 3$</annotation>\u0000 </semantics></math>, we show that all automorphisms preserve the coarse median structure induced, respectively, by the Davis complex and the Niblo–Reeves cubulation. As a consequence, automorphisms of these groups have nice fixed subgroups and satisfy Nielsen realisation.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12353","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Degenerations of \u0000 \u0000 k\u0000 $k$\u0000 -positive surface group representations","authors":"Jonas Beyrer,&nbsp;Beatrice Pozzetti","doi":"10.1112/topo.12352","DOIUrl":"https://doi.org/10.1112/topo.12352","url":null,"abstract":"<p>We introduce <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>-<i>positive representations</i>, a large class of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mtext>…</mtext>\u0000 <mo>,</mo>\u0000 <mi>k</mi>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation>$lbrace 1,ldots ,krbrace$</annotation>\u0000 </semantics></math>-Anosov surface group representations into <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>PGL</mi>\u0000 <mo>(</mo>\u0000 <mi>E</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathsf {PGL}(E)$</annotation>\u0000 </semantics></math> that share many features with Hitchin representations, and we study their degenerations: unless they are Hitchin, they can be deformed to non-discrete representations, but any limit is at least <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>k</mi>\u0000 <mo>−</mo>\u0000 <mn>3</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(k-3)$</annotation>\u0000 </semantics></math>-positive and irreducible limits are <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>k</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(k-1)$</annotation>\u0000 </semantics></math>-positive. A major ingredient, of independent interest, is a general limit theorem for positively ratioed representations.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Homeomorphism groups of 2-manifolds with the virtual Rokhlin property","authors":"Justin Lanier,&nbsp;Nicholas G. Vlamis","doi":"10.1112/topo.12354","DOIUrl":"https://doi.org/10.1112/topo.12354","url":null,"abstract":"<p>We introduce and motivate the definition of the virtual Rokhlin property for topological groups. We then classify the 2-manifolds whose homeomorphism groups have the virtual Rokhlin property. We also establish the analogous result for mapping class groups of 2-manifolds.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141968286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Applications of higher-dimensional Heegaard Floer homology to contact topology","authors":"Vincent Colin,&nbsp;Ko Honda,&nbsp;Yin Tian","doi":"10.1112/topo.12349","DOIUrl":"https://doi.org/10.1112/topo.12349","url":null,"abstract":"<p>The goal of this paper is to set up the general framework of higher-dimensional Heegaard Floer homology, define the contact class, and use it to give an obstruction to the Liouville fillability of a contact manifold and a sufficient condition for the Weinstein conjecture to hold. We discuss several classes of examples including those coming from analyzing a close cousin of symplectic Khovanov homology and the analog of the Plamenevskaya invariant of transverse links.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141597127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Characteristic cohomology II: Matrix singularities","authors":"James Damon","doi":"10.1112/topo.12330","DOIUrl":"https://doi.org/10.1112/topo.12330","url":null,"abstract":"&lt;p&gt;For a germ of a variety &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;⊂&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {V}, 0 subset mathbb {C}^N, 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, a singularity &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$mathcal {V}_0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of “type &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {V}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;”  is given by a germ &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;:&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$f_0: mathbb {C}^n, 0 rightarrow mathbb {C}^N, 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, which is transverse to &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mo&gt;∖&lt;/mo&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {V}setminus lbrace 0rbrace$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; in an appropriate sense, such that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {V}_0 = f_0^{-1}(mathcal {V})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In part I of this paper, we introduced for such singularities the Characteristic Cohomology for the Milnor fiber (for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 ","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141488838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}