Singularities and stable homotopy groups of spheres. II. (English) Zbl 1403.57025
This paper is a continuation of [C. Nagy et al., J. Singul. 17, 1–27 (2018; Zbl 1400.57026)], but can be read independently.
The authors investigate the cobordism groups of a particular class of Morin maps, namely the codimension \(1\) cooriented prim maps. A smooth map \(g:M^n\to \mathbb{R}^{n+1}\) is called prim if the kernel of the differential \(dg\) has rank at most one at all points of \(M\), and these kernels form a trivialised line bundle over the singular set. Such maps are so named because they are projections of immersions \(M^n\looparrowright \mathbb{R}^{n+2}\). The singularity types for such a map form an infinite sequence \(\Sigma^{1_r} = \Sigma^{1,\ldots,1,0}\) (with \(r\) digits 1), and a prim map \(g\) without \(\Sigma^{1_s}\)-singularities for \(s>r\) is called a prim \(\Sigma^{1_r}\)-map. It follows from the first author’s general results on classifying spaces for cobordism of maps with prescribed multi-singularities [A. Szűcs, Geom. Topol. 12, No. 4, 2379–2452 (2008; Zbl 1210.57028)] that there is a classifying space \(X\) for codimension \(1\) cooriented prim maps which is filtered by the classifying spaces \(X^r\) for prim \(\Sigma^{1_r}\)-maps. The authors offer an independent proof that \(X^r\) is homotopy equivalent to \(\Omega\Gamma\mathbb{C}P^{r+1}\), where \(\Gamma=\Omega^\infty S^\infty\) is the free infinite loop space functor, from which it follows that the cobordism group \(Prim\Sigma^{1_r}(n)\) of codimension \(1\) cooriented prim \(\Sigma^{1_r}\)-maps is isomorphic to the stable homotopy group \(\pi_{n+2}^S(\mathbb{C}P^{r+1})\). Furthermore, they identify the spectral sequence in homotopy arising from the mentioned filtration of \(X\) by the \(X^r\) with the spectral sequence in stable homotopy arising from the filtration of \(\mathbb{C}P^\infty\) by the \(\mathbb{C}P^r\), considered by Mosher in [R. E. Mosher, Topology 7, 179–193 (1968; Zbl 0172.25103)]. As a result of this identification, they glean information about the differentials, which describe how simpler singularity strata are incident to more complicated ones. In particular, the first non-vanishing differential lies in the image of the \(J\)-homomorphism, and can be calculated following Mosher by using the \(e\)-invariant of Adams. The authors do a valuable service to the community by explicating several of the arguments only sketched in Mosher’s paper [loc. cit.].
The authors investigate the cobordism groups of a particular class of Morin maps, namely the codimension \(1\) cooriented prim maps. A smooth map \(g:M^n\to \mathbb{R}^{n+1}\) is called prim if the kernel of the differential \(dg\) has rank at most one at all points of \(M\), and these kernels form a trivialised line bundle over the singular set. Such maps are so named because they are projections of immersions \(M^n\looparrowright \mathbb{R}^{n+2}\). The singularity types for such a map form an infinite sequence \(\Sigma^{1_r} = \Sigma^{1,\ldots,1,0}\) (with \(r\) digits 1), and a prim map \(g\) without \(\Sigma^{1_s}\)-singularities for \(s>r\) is called a prim \(\Sigma^{1_r}\)-map. It follows from the first author’s general results on classifying spaces for cobordism of maps with prescribed multi-singularities [A. Szűcs, Geom. Topol. 12, No. 4, 2379–2452 (2008; Zbl 1210.57028)] that there is a classifying space \(X\) for codimension \(1\) cooriented prim maps which is filtered by the classifying spaces \(X^r\) for prim \(\Sigma^{1_r}\)-maps. The authors offer an independent proof that \(X^r\) is homotopy equivalent to \(\Omega\Gamma\mathbb{C}P^{r+1}\), where \(\Gamma=\Omega^\infty S^\infty\) is the free infinite loop space functor, from which it follows that the cobordism group \(Prim\Sigma^{1_r}(n)\) of codimension \(1\) cooriented prim \(\Sigma^{1_r}\)-maps is isomorphic to the stable homotopy group \(\pi_{n+2}^S(\mathbb{C}P^{r+1})\). Furthermore, they identify the spectral sequence in homotopy arising from the mentioned filtration of \(X\) by the \(X^r\) with the spectral sequence in stable homotopy arising from the filtration of \(\mathbb{C}P^\infty\) by the \(\mathbb{C}P^r\), considered by Mosher in [R. E. Mosher, Topology 7, 179–193 (1968; Zbl 0172.25103)]. As a result of this identification, they glean information about the differentials, which describe how simpler singularity strata are incident to more complicated ones. In particular, the first non-vanishing differential lies in the image of the \(J\)-homomorphism, and can be calculated following Mosher by using the \(e\)-invariant of Adams. The authors do a valuable service to the community by explicating several of the arguments only sketched in Mosher’s paper [loc. cit.].
Reviewer: Mark Grant (Aberdeen)
MSC:
| 57R45 | Singularities of differentiable mappings in differential topology |
| 57R90 | Other types of cobordism |
| 55P42 | Stable homotopy theory, spectra |
| 55T25 | Generalized cohomology and spectral sequences in algebraic topology |
References:
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