Singularities and stable homotopy groups of spheres. II

Szűcs, András; Terpai, Tamás

doi:10.5427/jsing.2018.17b

Cited by 1 publication

(5 citation statements)

References 29 publications

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“…Finally, in analogy with the complex (codimension 2) case of [15] we obtain that the classifying space XSp r admits the representation XSp r = ΩΓHP r+1 . The so-called singularity spectral sequence (see [15] for details) in homotopy groups that arises from the sequence of fibrations XSp r−1 ⊂ XSp r ⊂ XSp r+1 ⊂ . .…”

Section: Quaternionic Prim Mapsmentioning

confidence: 76%

“…The cobordism group of such immersions is in one-to-one correspondence with the set of homotopy classes [SP, ΓHP ∞ ]; in particular taking P = S n+3 yields a group isomorphic to π n+4 (ΓHP ∞ ) = π s n+4 (HP ∞ ). Completely analogously to the codimension 2 oriented case (when a complex structure can be defined on the normal bundle, see [15]) we have that if the hyperplane projection of such an immersion is a Σ 1r -map (i.e. it has no singularity Σ 1 i for i > r), then the normal bundle of the immersion can be pulled back from HP r .…”

Section: Quaternionic Prim Mapsmentioning

confidence: 91%

“…Completely analogously to the codimension 2 oriented case (when a complex structure can be defined on the normal bundle, see [15]) we have that if the hyperplane projection of such an immersion is a Σ 1r -map (i.e. it has no singularity Σ 1 i for i > r), then the normal bundle of the immersion can be pulled back from HP r .…”

Section: Quaternionic Prim Mapsmentioning

confidence: 91%

“…Remark 1. In [15][Lemma 4] it is shown that when k = 2, we have X r = ΓCP r+1 . This is a special case of Theorem 1 as it follows from the next lemma.…”

Section: Quaternionic Prim Mapsmentioning

confidence: 99%

“…Let v denote the image of the upward-pointing vector ∂/∂z 5 in the normal bundle ofD 4 under the natural projection T R 8 → T R 8 /dŨ (TD 4 ) = νŨ . The classifying map sends v to a fixed section s of γ H 1 (that does not vanish on the given neighbourhood) and extends this bundle map to respect the quaternionic structure (see the proof of the second Claim in [15,Section 3]). In particular, the trivialization (s, is, js, ks) of γ H 1 over HP 0 that gives the generator of π s (0) = Z is pulled back to the trivialization (v, iv, jv, kv) of the normal bundle ofŨ , and the element in π s (3) ≈ Z 24 represented by this framed manifoldS 3 is the image of the differential d 1 2,6 : E 1 2,6 ∼ = Z → E 1 1,6 = π s (3) ∼ = Z 24 in the spectral sequence of subsection 2.2 evaluated on a generator of E 1 2,6 ∼ = Z.…”

Section: Proofs Of Theorem 1 and Theoremmentioning

confidence: 99%

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Classifying spaces for projections of immersions with controlled singularities

Szűcs

Terpai

2020

Acta Math. Hungar.

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We give an explicit simple construction for classifying spaces of maps obtained as hyperplane projections of immersions. We prove structure theorems for these classifying spaces. IntroductionDefinition. Let M n and P n+k be smooth manifolds and f : M n → P n+k a smooth (C ∞ -) map. f is called a corank 1 map if rank df x ≥ n − 1 for all x ∈ M n . A stable corank 1 map is called a Morin map.Definition. Given a Morin map f we say that x ∈ M n is a Σ 1r,0 -point if there exists a regular curve γ : (R, 0) → (M, x) going through x that has∂t r+1 (0) = 0. Morin [4] showed that for a fix r all Σ 1r,0 germs are left-right equivalent (A-equivalent) and that for r = s the Σ 1r,0 germs are not equivalent to Σ 1s,0 germs.Definition. Given a Morin map f : M n → P n+k we denote by Σ(f ) the set of its singular points and we denote by Σ 1r,0 (f ) the set of its Σ 1r,0 -points. Definition.A Morin map is called a Σ 1r -map if it has no Σ 1s,0 -points with s > r. Definition.A corank 1 map f : M n → P n+k equipped with a trivialization of its kernel line bundle is called a prim map. by the National Research, Development and Innovation Office NKFIH (OTKA) Grant K 120697. A. Szűcs and T. TerpaiNote that a prim map is the composition of an immersion g : M n P n+k × R 1 with the standard projection pr : P n+k × R 1 → P n+k . 1We denote by CobPrimΣ 1r (P ) the cobordism group of prim Σ 1r -maps in a fixed target manifold P , and we denote by CobΣ 1r (P ) the cobordism group of all Σ 1r -maps in a fixed target manifold P . For the standard definitions of these groups see [13]. Analogous groups can be defined for the case of cooriented maps or maps with a quaternionic normal structure; we denote them by Cob SO PrimΣ 1r (P ), Cob SO Σ 1r (P ) and Cob Sp PrimΣ 1r (P ), Cob Sp Σ 1r (P ), respectively.

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Section: Quaternionic Prim Mapsmentioning

confidence: 76%

Section: Quaternionic Prim Mapsmentioning

confidence: 91%

Section: Quaternionic Prim Mapsmentioning

confidence: 91%

“…Remark 1. In [15][Lemma 4] it is shown that when k = 2, we have X r = ΓCP r+1 . This is a special case of Theorem 1 as it follows from the next lemma.…”

Section: Quaternionic Prim Mapsmentioning

confidence: 99%

Section: Proofs Of Theorem 1 and Theoremmentioning

confidence: 99%

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