Classifying spaces for projections of immersions with controlled singularities
aa r X i v : . [ m a t h . G T ] F e b Classifying spaces for projections of immersionswith controlled singularities
Andr´as Sz˝ucs, Tam´as Terpai
Abstract
We give an explicit simple construction for classifying spaces ofmaps obtained as hyperplane projections of immersions. We provestructure theorems for these classifying spaces.
Definition.
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 map if rank df x ≥ n − 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 r , -point ifthere exists a regular curve γ : ( R , → ( M, x ) going through x that has ∂ r ( f ◦ γ ) ∂t r (0) = 0, but no such regular curve satisfies ∂ r +1 ( f ◦ γ ) ∂t r +1 (0) = 0.Morin [4] showed that for a fix r all Σ r , germs are left-right equivalent( A -equivalent) and that for r = s the Σ r , germs are not equivalent to Σ s , germs. Definition.
Given a Morin map f : M n → P n + k we denote by Σ( f ) the setof its singular points and we denote by Σ r , ( f ) the set of its Σ r , -points. Definition.
A Morin map is called a Σ r -map if it has no Σ s , -points with s > r . Definition.
A corank 1 map f : M n → P n + k equipped with a trivializationof its kernel line bundle is called a prim map. Acknowledgement: The authors were supported by the National Research, Develop-ment and Innovation Office NKFIH (OTKA) Grant NK 112735. TT was also supportedby the National Research, Development and Innovation Office NKFIH (OTKA) Grant K120697. A. Sz˝ucs and T. Terpai
Note that a prim map is the composition of an immersion g : M n P n + k × R with the standard projection pr : P n + k × R → P n + k . We denote by CobPrimΣ r ( P ) the cobordism group of prim Σ r -mapsin a fixed target manifold P , and we denote by CobΣ r ( P ) the cobordismgroup of all Σ r -maps in a fixed target manifold P . For the standard def-initions of these groups see [13]. Analogous groups can be defined for thecase of cooriented maps or maps with a quaternionic normal structure; wedenote them by Cob SO PrimΣ r ( P ), Cob SO Σ r ( P ) and Cob Sp PrimΣ r ( P ),Cob Sp Σ r ( P ), respectively.Whenever there is a (prim) Σ r -map f : M → P and a smooth map g : P ′ → P a standard pullback diagram arises: M ′ g ∗ f / / ❴❴❴ (cid:15) (cid:15) ✤✤✤ P ′ g (cid:15) (cid:15) M f / / P If the map g is transverse to all the submanifolds f (cid:0) Σ s , ( f ) (cid:1) for s =0 , . . . , r , then the map g ∗ f is a (prim) Σ r -map as well. If the map g isnot transverse to the submanifolds Σ s ( f ), one can still choose an approxi-mating map ˜ g close to g that is transverse and obtain a (prim) Σ r -map ˜ g ∗ f ;this map is not unique, but any two choices of such approximating maps —˜ g and ˜ g , say — can be deformed into one another by a homotopy G thatis itself transverse to the submanifolds Σ s ( f ), hence ˜ g ∗ f and ˜ g ∗ f are con-nected by the cobordism G ∗ f and represent the same element in the (prim)Σ r -cobordism group of P . It is easy to check that sending f to the pullbackmap f ˜ g ∗ f induced by a suitable approximation of g on the cobordismgroup of P we obtain a contravariant functor from the category of smoothmanifolds and smooth maps to the category of groups and homomorphisms.A similar functor can be defined using CobPrimΣ r ( · ) instead of CobΣ r ( · ).There exist (homotopically unique) spaces X r and X r that represent thefunctors P −→ CobΣ r ( P ) and P −→ CobPrimΣ r ( P ) The word prim is an abbreviation of “ pr ojection of im mersion”. lassifying spaces for projections of immersions (see [10]), in particularCobΣ r ( P ) = [ P, X r ] andCobPrimΣ r ( P ) = [ P, X r ]for any compact manifold P . We call the spaces X r and X r the classifyingspaces for Σ r -maps and prim Σ r -maps, respectively. This type of clas-sifying spaces in a more general setup has been explicitly constructed andinvestigated earlier, see [12], [11]; in [7] and [13] two significantly differentexplicit descriptions of the classifying space for a more general class of sin-gular maps are given, and in [16] a homotopy theoretic connection betweenthose constructions is established. Again, analogues for oriented maps andquaternionic maps can be defined, we denote them by X SO r , X SO r , X Sp r and X Sp r ; in what follows, we will omit the distinguishing upper indices whenthe argument works for each case.In the present paper we give a simple construction for the spaces X r and prove structure theorems for them. As a byproduct we get an explicitdescription of some elements of stable homotopy groups of spheres via localforms of Morin maps. X r , the classifying space of co-bordisms of prim Σ r -maps Notation:
Let γ O k +1 → BO ( k + 1) denote the universal ( k + 1)-dimensionalvector bundle, let γ SO k +1 → BSO ( k +1) denote the universal ( k +1)-dimensio-nal oriented vector bundle and denote by γ Sp k +1 → BSp ( k + 1) the univer-sal 4( k + 1)-dimensional quaternionic vector bundle. We denote by γ k +1 one of these bundles, with the implication that our arguments apply toeach case. Let S = S (( r + 1) γ k +1 ) be the sphere bundle of ( r + 1) γ k +1 = r +1 z }| { γ k +1 ⊕ · · · ⊕ γ k +1 , with pr S : S → B ( k + 1) denoting the projection on B ( k + 1), which is either BO ( k + 1), BSO ( k + 1), or BSp ( k + 1). Definethe bundle ζ S to be the pullback pr ∗ S γ k +1 . The Thom space of ζ S will bedenoted by T ζ S , and ΩΓ T ζ S is the space Ω ∞ +1 S ∞ T ζ S = lim q →∞ Ω q +1 S q T ζ S . Theorem 1. X r = ΩΓ T ζ S . In order to apply Brown’s theorem directly, one has to extend these functors to arbi-trary simplicial complexes (not only manifolds). This is done in [13].
A. Sz˝ucs and T. Terpai
From here onward, the symbol ∼ = Q stands for rational homotopy equiv-alence. Theorem 2. a) There is a fibration X r Ω Γ T (( r +2) γ k +1 ) −−−−−−−−−−−→ ΩΓ T γ k +1 .b) If k is odd, then ΩΓ T γ SO k +1 ∼ = Q X SO r × ΩΓ T (cid:0) ( r + 2) γ SO k +1 (cid:1) c) If k is even, then X SO r ∼ = Q Ω Γ T (cid:0) ( r + 2) γ SO k +1 (cid:1) × ΩΓ T γ SO k +1 The proofs are postponed to Section 3.An important application of Theorem 2 is that it allows the calculationof the ranks of the groups Cob SO PrimΣ r ( P ) ∼ = [ P, X SO r ] for arbitrary targetmanifolds P . First recall that the H -space X SO r rationally splits as a productof Eilenberg-MacLane spaces X SO r ∼ = Q ∞ Y j =1 K ( π j ( X SO r ) ⊗ Q ; j ) . This implies that for every P we have(2.1) [ P, X SO r ] ⊗ Q ∼ = ∞ M j =1 [ P, K ( π j ( X SO r ) ⊗ Q ; j )] ∼ = ∼ = ∞ M j =1 H j ( P ; Z ) ⊗ π j ( X SO r ) ⊗ Q and we only need to calculate the ranks of the homotopy groups of X SO r .When k is odd, Theorem 2 yieldsrank π j ( X SO r ) = rank π j (ΩΓ T γ SO k +1 ) − rank π j (cid:0) ΩΓ T (cid:0) ( r + 2) γ SO k +1 (cid:1)(cid:1) == rank π sj +1 ( T γ SO k +1 ) − rank π sj +1 (cid:0) T (cid:0) ( r + 2) γ SO k +1 (cid:1)(cid:1) == rank H j +1 ( T γ SO k +1 ; Z ) − rank H j +1 (cid:0) T (cid:0) ( r + 2) γ SO k +1 (cid:1) ; Z (cid:1) == rank H j − k ( BSO ( k + 1); Z ) −− rank H j +1 − ( r +2)( k +1) ( BSO ( k + 1); Z ); lassifying spaces for projections of immersions k is even we getrank π j ( X SO r ) = rank π j (cid:0) Ω Γ T (cid:0) ( r + 2) γ SO k +1 (cid:1)(cid:1) + rank π j (ΩΓ T γ SO k +1 ) == rank π sj +2 (cid:0) T (cid:0) ( r + 2) γ SO k +1 (cid:1)(cid:1) + rank π sj +1 ( T γ SO k +1 ) == rank H j +2 (cid:0) T (cid:0) ( r + 2) γ SO k +1 (cid:1) ; Z (cid:1) + rank π sj +1 ( T γ SO k +1 ; Z ) == rank H j +2 − ( r +2)( k +1) ( BSO ( k + 1); Z )++ rank H j − k ( BSO ( k + 1); Z ) . Substituting the ranks of the homology groups H ∗ ( BSO ( k + 1); Z ) into theformula (2.1) we obtain the following expressions: Corollary.
Denote by p ≤ t ( m ) the number of partitions of m into positiveintegers between and t (in particular, p ≤ t ( m ) = 0 whenever m is not anonnegative integer).a) If k is odd, rank Cob SO PrimΣ r ( P ) = ∞ X j =1 rank H j ( P, Z ) ×× (cid:18) p ≤ k − (cid:18) j − k (cid:19) + p ≤ k − (cid:18) j − k − (cid:19) −− p ≤ k − (cid:18) j + 1 − ( r + 2)( k + 1)4 (cid:19) + p ≤ k − (cid:18) j − k − ( r + 2)( k + 1)4 (cid:19)(cid:19) . b) If k is even, rank Cob SO PrimΣ r ( P ) = ∞ X j =1 rank H j ( P, Z ) ×× (cid:18) p ≤ k (cid:18) j − k (cid:19) + p ≤ k (cid:18) j + 2 − ( r + 2)( k + 1)4 (cid:19)(cid:19) . Analogous computations can be performed to obtain the ranks of theprim Σ r cobordism groups in the non-oriented and in the quaternioniccases as well. Denote by X ∞ the classifying space of all prim maps (of codimension k ).Clearly X ∞ = lim r →∞ X r . By considering the immersion lifts of the involvedmaps it is easy to see that X ∞ = ΩΓ T γ k +1 . A. Sz˝ucs and T. Terpai
Lemma 3.
The homotopy exact sequence of the fibration in Theorem 2 a ) can be identified with the homotopy exact sequence of the pair ( X ∞ , X r ) .Proof. We first construct a map α : π n + k +1 ( X ∞ , X r ) → π n + k (cid:0) Ω Γ T (( r + 2) γ k +1 ) (cid:1) == π n + k +1 (ΩΓ T (( r + 2) γ k +1 )) . The relative homotopy group π n + k +1 ( X ∞ , X r ) can clearly be identifiedwith the relative cobordism group of those prim maps F : ( N n +1 , ∂N ) → ( R n + k +1+ , R n + k ) that possess the property of F | ∂N : ∂N → R n + k being aΣ r -map. The map α associates to the cobordism class of F the cobordismclass of its Σ r +1 -points equipped with the normal structure of its immersionlift, which is a splitting of the normal bundle into ( r + 2) isomorphic bun-dles. This kind of maps is classified by the space ΩΓ T (( r + 2) γ k +1 ). Thuswe obtain a chain map of the homotopy exact sequence of the pair ( X ∞ , X r )to that of the fibration of Theorem 2 a ). The five lemma implies that α isan isomorphism.Theorem 2 b ) and c ) states that this exact sequence splits rationally. Toelaborate, if k is odd then the sequence0 → π n + k ( X r ) → π n + k ( X ∞ ) → π n + k ( X ∞ , X r ) → r -mapexactly if its Σ r +1 -singularity stratum is rationally null-cobordant.If k is even, then the same is true for the sequence0 → π n + k +1 ( X ∞ , X r ) → π n + k ( X r ) → π n + k ( X ∞ ) → r -map. Furthermore a Σ r -map that is rationally null-cobordant as anarbitrary prim map is determined up to rational cobordism by the ratio-nal cobordism class of the Σ r +1 -stratum of any prim map that it boundsrationally. Given an ( n + 3)-dimensional manifold P n +3 let us consider immersionsof closed cooriented n -dimensional manifolds immersed in P × R with a lassifying spaces for projections of immersions Sp (1) ∼ = Spin (3) ∼ = S , the group of unit quaternions.The cobordism group of such immersions is in one-to-one correspondencewith the set of homotopy classes [ SP, Γ H P ∞ ]; in particular taking P = S n +3 yields a group isomorphic to π n +4 (Γ H P ∞ ) = π sn +4 ( H P ∞ ) . Completely analogously to the codimension 2 oriented case (when a com-plex structure can be defined on the normal bundle, see [15]) we have thatif the hyperplane projection of such an immersion is a Σ r -map (i.e. it hasno singularity Σ i for i > r ), then the normal bundle of the immersion canbe pulled back from H P r . The inverse is also true up to regular homo-topy: if the immersion has its normal bundle pulled back from the canonicalquaternionic line bundle over H P r , then it can be deformed by a regular ho-motopy into an immersion such that its hyperplane projection is a Σ r -map.We shall call such prim maps quaternionic Σ r -prim maps .The cobordism group of such maps into P can be defined in a standardway and will be denoted by Cob Sp PrimΣ r ( P ).Let ¯ X Spr denote the classifying space of these cobordism groups, so thatit satisfies Cob Sp PrimΣ r ( P ) = [ ˙ P , ¯ X Spr ] ∗ Here ˙ P denotes the one-point compactification of P (if P itself is com-pact, then this is the disjoint union of P and an extra point); [ − , − ] ∗ denotesthe set of pointed homotopy classes.Finally, in analogy with the complex (codimension 2) case of [15] weobtain that the classifying space ¯ X Spr admits the representation¯ X Spr = ΩΓ H P r +1 . The so-called singularity spectral sequence (see [15] for details) in homo-topy groups that arises from the sequence of fibrations¯ X Spr − ⊂ ¯ X Spr ⊂ ¯ X Spr +1 ⊂ . . . coincides (after a shift of the indices) with the spectral sequence in stablehomotopy groups of the filtration H P ⊂ H P ⊂ . . . H P r ⊂ · · · ⊂ H P ∞ A. Sz˝ucs and T. Terpai
The first page of this spectral sequence is E p,q = π sp + q ( H P p / H P p − ) = π sp + q ( S p ) = π s ( q − p ) q = 10 π s (7) ∼ = Z π s (4) = 0 Z q = 9 π s (6) ∼ = Z h ν i π s (3) ∼ = Z o o o o Z o o k k ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ q = 8 0 0 q = 7 0 0 q = 6 π s (3) ∼ = Z Z o o o o q = 5 π s (2) ∼ = Z q = 4 π s (1) ∼ = Z q = 3 Z p = 1 p = 2 p = 3The only non-finite groups among the groups E p,q are hence those onthe line q = 3 p , these are all Z . Lemma 4.
The group E ∞ p, p ∼ = Z considered as a subgroup of E p, p ∼ = Z hasindex equal to the order of the cokernel of the stable Hurewicz homomorphism π sp + q ( H P ∞ ) → H p + q ( H P ∞ ) .Proof. Consider the degenerate homological spectral sequence starting with H E p,q = H p + q ( H P p / H P p − ) (which is Z if q = 3 p and 0 otherwise) for thesame filtration and the map from the spectral sequence E ∗ p,q into it induced lassifying spaces for projections of immersions E p, p ∼ = H E p, p , both isomorphic to Z . On the final page we have E ∞ p, p = Z identified with the free part of π s p ( H P ∞ ), and it is mapped into H E ∞ p, p = H p ( H P ∞ ) = Z , the imageof this homomorphism being the same as the image of the stable Hurewiczhomomorphism. Corollary.
The product of the orders of the images of the differentials d rp, p : Z = E rp, p → E rp − r, p + r − (taken for all r from to infinity) is equal tothe index of the image of the stable Hurewicz homomorphism π s p ( H P ∞ ) → H p ( H P ∞ ) . Segal [9] has determined this index:
Theorem 5. [9, Theorem 1.1.] The image of π s p ( H P ∞ ) → H p ( H P ∞ ) is h ( p ) · Z , where • h ( p ) = (2 p )! for p even and • h ( p ) = (2 p )! / for p odd. It follows that all the differentials of the fragment of the spectral sequenceseen on the diagram are epimorphic modulo the 2-primary torsion part, andthe differential d , : E , → E , is (truly) epimorphic. In particular weobtain that the boundaries of the normal forms of the Morin maps of typeΣ , and Σ , , in codimension 3 give a generator of the stable homotopygroup of spheres π s (3) and a generator of the odd torsion of the stablehomotopy group of spheres π s (7), respectively. In the Appendix we describethe first of these classes in more detail.We also get that the torsion parts of π si ( H P ∞ ) in the range i ≤ Sp PrimΣ ( R n +3 ) for n ≤ n ≡ X Spr : Theorem 6. X Spr ∼ = Q S × S × · · · × S r +3 . Proof.
We repeat the argument that determines the rational homotopy typeof C P m (which we learned from D. Crowley); we show by induction that thestable rational homotopy type of H P m is that of S ∨ S ∨ · · · ∨ S m . Bythe induction hypothesis H P m − ∼ = Q S ∨ S ∨ · · · ∨ S m − and the stable0 A. Sz˝ucs and T. Terpai homotopy class of the attaching map of the top dimensional cell of H P m isdefined by stable maps from S m − to S i for i = 1 , , . . . , m −
1, and allthese stable maps are rationally trivial (they have finite order).Consequently X Spr = ΩΓ H P r +1 ∼ = Q ΩΓ( S ∨ S ∨ · · · ∨ S r +4 ) ∼ = Γ S × Γ S × · · · × Γ S r +3 ∼ = Q S × S × · · · × S r +3 (we used that ΩΓ S = Γ;Γ( A ∨ B ) = Γ A × Γ B ; and Γ S i ∼ = Q S i ). Remark 1.
In [15][Lemma 4] it is shown that when k = 2, we have X r =Γ C P r +1 . This is a special case of Theorem 1 as it follows from the nextlemma. Recall that if B ( k + 1) = BSO (2), then ζ S = π ∗ r γ SO2 , where γ SO2 is the universal oriented vector bundle of rank 2 and π r : S (cid:0) ( r + 1) γ SO2 (cid:1) → BSO (2) is the sphere bundle of the vector bundle γ SO2 ⊕ · · · ⊕ γ SO2 (with( r + 1) summands). We denote by γ C the universal complex line bundle(over C P ∞ ). Lemma 7.
The vector bundle ζ S → S (cid:0) ( r + 1) γ SO2 (cid:1) and the complex linebundle γ C | C P r → C P r are homotopically equivalent in the sense that thereis a homotopy equivalence f : C P r → S (( r + 1) γ SO2 ) such that f ∗ ζ S is iso-morphic as an oriented rank real vector bundle to the tautological complexline bundle over C P r .Proof. It is well-known that γ C can be identified with γ SO2 , and the tauto-logical complex line bundle over C P r is the restriction γ C | C P r .Consider the space S ∞ × S ( C r +1 ) × C and the natural diagonal S -action on it: for g ∈ S and ( x, y, z ) ∈ S ∞ × S ( C r +1 ) × C set g ( x, y, z ) =( gx, gy, gz ). The subspace S ∞ × S ( C r +1 ) × { } is invariant under thisaction; the corresponding orbit space is S ∞ × S S ( C r +1 ). Regarding this orbitspace as an S ( C r +1 )-bundle over S ∞ /S we identify it with S (cid:0) ( r + 1) γ C (cid:1) .Regarding the same orbit space as an S ∞ -bundle over S ( C r +1 ) /S = C P r we get that it is homotopically equivalent to C P r . The obtained homotopyequivalence between C P r and S (cid:0) ( r + 1) γ C (cid:1) takes the tautological complexline bundle to (pullback of) the tautological complex line bundle since itextends to the entire orbit space ( S ∞ × S ( C r +1 ) × C ) /S , which is the totalspace of these bundles; this finishes the proof of the lemma.Similarly ζ SpS → S (cid:16) ( r + 1) γ Sp (cid:17) is homotopically equivalent to the vectorbundle γ H | H P r → H P r . This combined with Theorem 1 gives that X Spr isΩΓ H P r +1 . Here S ( C r +1 ) is considered to be the space of unit length complex vectors in C r +1 . lassifying spaces for projections of immersions Proof of Theorem 1.
Given a generic immersion g : M n P n + k × R wefirst produce sections s , s , . . . of the normal bundle ν g such that Σ j ( f ) = ∩ ji =1 s − i (0), where f = pr ◦ g , the map pr : P × R → P being the projection.The (positive) basis vector of R defines a constant vector field on P × R that we call (upward directed) vertical and denote by ↑ . Project ↑ into thenormal bundle ν g (considered as the quotient bundle T ( P × R ) /dg ( T M ),and denote the obtained section by s . Note that the singularity set Σ( f )of f is precisely s − (0), the zero set of the section s .For a generic map f the set Σ( f ) is a manifold of codimension k + 1.Denote by ν the normal bundle of Σ( f ) in M . Note that ν ∼ = ν g | Σ( f ) : thetangent space of the section s at the points of Σ( f ) is the graph of a linearisomorphism β : ν → ν g | Σ( f ) .In order to produce the section s we first define a section z of ν byprojecting ↑ into ν at the points of Σ( f ) (where ↑ ∈ dg ( T M ), hence thedefinition makes sense). Applying the isomorphism β we get a section s ′ = β ◦ z of ν g | Σ( f ) . The section s is defined as an arbitrary (continuous)extension of s ′ to the entire ν g . Clearly Σ ( f ) is the zero set of z , henceΣ ( f ) = s − (0) ∩ s − (0). We continue in the same fashion, producingsections s , . . . , s r +1 such that Σ j ( f ) = ∩ ji =1 s − i (0). In particular if f is aΣ r -map, then ∩ r +1 i =1 s − i (0) = ∅ .Note that the sections s , . . . , s r are not unique, but each one is chosenuniquely up to a contractible choice. The difference of any two possiblechoices of s is an arbitrary section of ν g that vanishes on s − (0). Thedifference of any two possible choices of s is arbitrary section of the normalbundle ν g that vanishes on s − (0) ∩ s − (0) etc. Hence the collection ofthese sections defines a homotopically unique section α of the sphere bundle p Sr ( g ) : S (( r + 1) ν g ) → M .Let G k +1 denote the infinite Grassmann manifold of all k + 1-planes in R ∞ and let ϕ : M → G k +1 be the map that induces ν g from the universalbundle γ k +1 , furthermore let Φ : ν g → γ k +1 be the corresponding fiberwiseisomorphism. Φ induces a map Φ rS : S (( r +1) ν g ) → S (( r +1) ν k +1 ). Consider2 A. Sz˝ucs and T. Terpai the following diagram: ζ g / / (cid:18) (cid:18) % % ❑❑❑❑❑❑❑❑❑❑❑ ζ S (cid:15) (cid:15) x x ♣♣♣♣♣♣♣♣♣♣♣♣ S (( r + 1) ν g ) Φ rS / / p Sr ( g ) (cid:21) (cid:21) S (( r + 1) γ k +1 ) (cid:15) (cid:15) M f α r =Φ rS ◦ α ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ ϕ / / α T T G k +1 ν g rrrrrrrrrrrr Φ / / A R R γ k +1 g g ❖❖❖❖❖❖❖❖❖❖❖❖ ( ∗ r )(since ζ g = p Sr ( g ) ∗ ( ν g ) and α is a section of the bundle map p Sr ( g ), the map α can be lifted into a vector bundle map A : ν g → ζ g ). Its commutativitygives us that f α r ∗ ζ S = ν g . Thus we have obtained the proof of the followinglemma: Lemma 8. If g is a generic immersion such that f = pr ◦ g is a Σ r -map, then the normal bundle ν g can be induced by f α r from ζ S , which inturn is the pullback of γ k +1 → G k +1 to S (( r + 1) γ k +1 ) by the projection S (( r + 1) γ k +1 ) → G k +1 . By the remark above about the contractible choices of the sections s , . . . , s r +1 we have seen that α is homotopically unique and therefore the map f α r is also homotopically unique. Hence ν g can be pulled back from ζ S in ahomotopically well-defined way.Applying the Pontryagin-Thom construction to the diagram ( ∗ r ) we con-struct a map( ∗∗ ) CobPrimΣ r ( P ) → [ P, ΩΓ T ζ S ]This map arises as follows: the cobordism class of an immersion g : M P × R gives a map SP → Γ T ζ S . Hence the map f = pr ◦ g gives a map P → ΩΓ T ζ S ; this map is homotopically unique and its homotopy class is thesame for any representative of the cobordism class of g ; hence also for thatof f . Let us denote the classifying space for prim Σ r -cobordism by X r . Themap ( ∗∗ ) is induced by a (homotopically unique) map θ r : X r → ΩΓ T ζ S between the classifying spaces. In [13] we have shown that there is a fibration X r − → X r p r → Γ S r T (( r + 1) γ k ) and the map p r induces the forgetful mapthat sends the cobordism class of a Σ r -map to the cobordism class of theimmersed top singularity stratum. Note that the base space can be rewritten lassifying spaces for projections of immersions S r T (( r + 1) γ k ) = ΩΓ T (cid:0) ( r + 1)( γ k ⊕ ε ) (cid:1) . Hence there is a long exactsequence . . . → CobPrimΣ r − ( SP ) → CobPrimΣ r ( SP ) → [ SP, ΩΓ T ( r + 1) γ k +1 ] →→ CobPrimΣ r − ( P ) → CobPrimΣ r ( P ) → [ P, ΩΓ T ( r + 1) γ k +1 ] → ∗∗ ). Claim 1.
There is a cofibration
T ζ S | S ( rγ k +1 ) ⊂ T ζ S → T ( r + 1)( γ k ⊕ ε ) . To see that Claim 1 holds, we utilize a trivial lemma:
Lemma 9.
Let N be a manifold, A ⊂ N be a submanifold with a tubularneighbourhood V and normal bundle ν , and let ξ be a vector bundle over N .Denote by ξ A and ξ N \ V the restrictions of ξ to A and N \ V , respectively.Then there is a cofibration of Thom spaces T ξ N \ V → T ξ → T ( ν ⊕ ξ A ) . Claim 1 follows by applying Lemma 9 to N = S (( r + 1) γ k +1 ), A = S ( γ k +1 ), ξ = ζ S with V = S (( r + 1) γ k +1 ) \ S ( rγ k +1 ) and ν = r ( γ k ⊕ ε ).Applying the functor ΩΓ to the cofibration of Claim 1 we obtain a fibra-tion ΩΓ T ζ S ΩΓ T ζ S | S ( rγk +1) −−−−−−−−−−→ ΩΓ T (cid:0) ( r + 1)( γ k ⊕ ε ) (cid:1) . There are also natural maps θ r − : X r − → ΩΓ T ζ S | S ( rγ k +1 ) and θ r : X r → ΩΓ T ζ S that correspond to the natural transformation of functorsCobPrimΣ r − ( · ) → [ · , ΩΓ T ζ S | S ( rγ k +1 ) ] andCobPrimΣ r ( · ) → [ · , ΩΓ T ζ S ]4 A. Sz˝ucs and T. Terpai given by ( ∗∗ ). We hence obtain the following map of fibrations: X r − θ r − / / (cid:15) (cid:15) ΩΓ T ζ S | S ( rγ k +1 ) (cid:15) (cid:15) X r θ r / / (cid:15) (cid:15) ΩΓ T ζ S (cid:15) (cid:15) ΩΓ T (cid:0) ( r + 1)( γ k ⊕ ε ) (cid:1) ∼ = / / ΩΓ T (cid:0) ( r + 1)( γ k ⊕ ε ) (cid:1) We show that this diagram commutes homotopically by proving that thediagram of the induced natural maps between the corresponding functorscommutes. The commutativity of the top square is obvious. In the bottomsquare, the normal structure encoded in ( r + 1)( γ k ⊕ ε ) on the left is thatof the splitting of the normal bundle of Σ r , into r + 1 bundles canonicallyisomorphic to the restriction of the normal bundle of the immersion lift, whilethe normal structure on the right encodes the common zero sets of the r + 1sections s j . While the structure on the right may be twisted with respectto that on the left by isomorphisms provided by the sections themselves,one can still recover one normal structure from the other uniquely up tohomotopy. Proof of Theorem 2. a)
Consider the vector bundle γ k +1 pulled back to thedisc bundle D (( r + 1) γ k +1 ) by the projection. Note that the Thom spaceof this bundle is homotopy equivalent to T γ k +1 , while the total space ofits disc bundle is that of the disc bundle D (( r + 2) γ k +1 ). Recall that afteridentifying γ k +1 with this pullback to the disc bundle the vector bundle ζ S is the restriction γ k +1 | S (( r +1) γ k +1 ) , and notice that T ζ S → T γ k +1 → T (( r + 2) γ k +1 ) is a cofibration by Lemma 9. By applying the functor ΩΓto it, we obtain the fibrationΩΓ T ζ S → ΩΓ T γ k +1 → ΩΓ T (( r + 2) γ k +1 ) . It is well-known (see e.g. [5]) that when one turns the inclusion ΩΓ
T ζ S → ΩΓ T γ k +1 of the fiber into a fibration, its fiber will be the loop space ofthe base, Ω Γ T (( r + 2) γ k +1 ). This fibration ΩΓ T ζ S = X r Ω Γ T (( r +2) γ k +1 ) −−−−−−−−−−−→ ΩΓ T γ k +1 is the one stated by Theorem 2 a ). b) When k is odd and we are in the oriented setting, the vector bundle γ SOk +1 has a nonvanishing Euler class and so does the vector bundle ( r +1) γ SOk +1 as well, hence (using the Gysin sequence) we obtain that the projection lassifying spaces for projections of immersions pr : S (( r + 1) γ SOk +1 ) → BSO ( k + 1) induces an epimorphism in cohomol-ogy with rational coefficients (here we use the fact that H ∗ ( BSO ( k + 1))has no zero divisors as a subring of the polynomial ring H ∗ ( BT ) with T the maximal torus of BSO ( k + 1)). Consequently so does the induced map T ζ S → T γ
SOk +1 of Thom spaces, therefore the cofibration T ζ S → T γ
SOk +1 → T (( r + 2) γ SOk +1 ) splits homologically and so the long exact sequence of the fi-bration Γ T ζ S → Γ T γ
SOk +1 → Γ T (( r + 2) γ SOk +1 ) in homotopy – which coincideswith the long exact sequence of the original cofibration in stable homo-topy – splits rationally as well. Since all the involved spaces are H-spacesand thus rationally products of Eilenberg-MacLane spaces, this implies thatΩΓ T γ
SOk +1 ∼ = Q ΩΓ T ζ S × ΩΓ T (cid:0) ( r + 2) γ SOk +1 (cid:1) = X SOr × Γ T (cid:0) ( r + 2) γ SOk +1 (cid:1) asclaimed. c) When k is even (and we are still in the oriented setting), the vectorbundle γ SOk +1 has vanishing Euler class and so does the vector bundle ( r +1) γ SOk +1 as well, hence the projection pr : S (( r + 1) γ SOk +1 ) → BSO ( k ) inducesa monomorphism in cohomology with rational coefficients. Consequentlyso does the induced map T ζ S → T γ
SOk +1 of Thom spaces. Extending thecofibration T ζ S → T γ
SOk +1 → T (( r + 2) γ SOk +1 ) to form the Puppe sequence T ζ S → T γ
SOk +1 → T (( r + 2) γ SOk +1 ) →→ ST ζ S → ST γ
SOk +1 → ST (( r + 2) γ SOk +1 ) → . . . we observe that the induced map H ∗ (cid:0) ST (cid:0) ( r + 2) γ SOk +1 (cid:1)(cid:1) → H ∗ (cid:0) ST γ
SOk +1 (cid:1) is also zero, therefore the cofibration T (cid:0) ( r + 2) γ SOk +1 (cid:1) → ST ζ S → ST γ
SOk +1 splits homologically and so the long exact sequence of the fibrationΩ Γ T (cid:0) ( r + 2) γ SOk +1 (cid:1) → Ω Γ ST ζ S → Ω Γ ST γ
SOk +1 in homotopy splits rationally as well. But since Ω Γ ST ζ S = ΩΓ T ζ S andΩ Γ ST γ
SOk +1 = ΩΓ T γ k +1 are H-spaces, we have X r = ΩΓ T ζ S ∼ = Q ΩΓ T γ k +1 × Ω Γ T (( r + 2) γ k +1 )as claimed. π s (3) via singularitytheory Consider the normal form of a Whitney umbrella map U : R → R givenby the coordinate functions( x, t , t , t ) ( y , y , y , z , z , z , z )6 A. Sz˝ucs and T. Terpai y m = t m m = 1 , , z m = t m x m = 1 , , z = x By adding the eighth coordinate function z = x we lift this map to anembedding ˜ U : R → R . Consider the restriction of ˜ U to ˜ D = U − ( D )and denote by ˜ S the boundary of ˜ D . Note that ˜ D is diffeomorphic tothe standard ball D . The map ˜ U is an embedding, hence ˜ U ( ˜ D ) is anembedded ball in R . Its normal bundle admits a homotopically uniquequaternionic line bundle structure. The direction of the added 8th coor-dinate line ( z ) is not tangent to ˜ D at the points of ˜ S (actually at anypoint of ˜ D except for the origin). Hence the classifying map sends thissubset to a ball neighbourhood of H P ⊂ H P and induces the normal bun-dle of ˜ U from γ H in the following manner. Let v denote the image of theupward-pointing vector ∂/∂z in the normal bundle of ˜ D under the naturalprojection T R → T R /d ˜ U ( T ˜ D ) = ν ˜ U . The classifying map sends v toa fixed section s of γ H (that does not vanish on the given neighbourhood)and extends this bundle map to respect the quaternionic structure (see theproof of the second Claim in [15, Section 3]). In particular, the trivialization( s, is, js, ks ) of γ H over H P that gives the generator of π s (0) = Z is pulledback to the trivialization ( v, iv, jv, kv ) of the normal bundle of ˜ U , and theelement in π s (3) ≈ Z represented by this framed manifold ˜ S is the imageof the differential d , : E , ∼ = Z → E , = π s (3) ∼ = Z in the spectralsequence of subsection 2.2 evaluated on a generator of E , ∼ = Z . Since thedifferential d , is surjective, it sends the generator of E , to a generator of E , = π s (3) ∼ = Z . References [1] M.G. Barratt, P.J. Eccles: Γ + -structures I., Topology 13 (1974), pp.25–45.[2] A. Hatcher: Algebraic Topology, http://pi.math.cornell.edu/~hatcher/AT/ATch4.4.pdf [3] J.P. May: The geometry of iterated loop spaces, Lecture Notes inMathematics 271, Springer, 1972[4] B. Morin: Formes canoniques des singularit´es d’une applicationdiff´erentiable, Comptes Rendus (1965), pp. 5662–5665, pp. 6502–6506. lassifying spaces for projections of immersions
J. Sing. (2008), 1–28.[7] R. Rim´anyi, A. Sz˝ucs: Pontrjagin–Thom-type construction for mapswith singularities, Topology
37 (6) , pp. 1177–1191.[8] C. Rourke, B. Sanderson: The Compression Theorem I,
Geometry &Topology (2001), pp. 399–429.[9] D. M. Segal: On the stable homotopy of quaternionic and complexprojective spaces, Proc. Amer. Math. Soc. (1970), pp. 838–841.[10] R. M. Switzer: Algebraic topology – homotopy and homology, reprintof the 1975 original, Classics in Mathematics, Springer-Verlag, Berlin,2002[11] A. Sz˝ucs: Cobordism of maps with simplest singularities, TopologySymposium, Siegen
Proc. Sympos., Univ. Siegen , Siegen, 1979),pp. 223–244., Lecture Notes in Math. , Springer, Berlin, 1980[12] A. Sz˝ucs: Analogue of the Thom space for mappings with singularityof type Σ , (Russian) Mat. SB. (N.S) (150) (1979) no.3. pp. 433–456., 478., English translation: Math. USSR-Sb. 36 (1979) no. 3. pp.405–426. (1980)[13] A. Sz˝ucs: Cobordism of singular maps, Geom. Topol. , no. 4 (2008),2379–2452.[14] E. Szab´o, A. Sz˝ucs, T. Terpai: On bordism and cobordism groups ofMorin maps, Journal of Singularities (2010), pp. 134–145.[15] A. Sz˝ucs, T. Terpai: Singularities and stable homotopy groups ofspheres II, J. Sing.17