ON PIERCE-LIKE IDEMPOTENTS AND HOPF INVARIANTS

Given a set K with cardinality ‖ K ‖ = n , a wedge 
decomposition of a space Y indexed by K , and a cogroup A , 
the homotopy group G = [ A , Y ] is shown, by using Pierce-like 
idempotents, to have a direct sum decomposition indexed by 
 P ( K ) − { ϕ } which is strictly functorial if G is abelian. 
Given a class ρ : X → Y , there is a Hopf invariant 
 HI ρ on [ A , Y ] which extends Hopf's definition when ρ is a comultiplication. Then HI = HI ρ is a functorial sum of HI L over L ⊂ K , ‖ L ‖ ≥ 2 . Each HI L is a 
functorial composition of four functors, the first depending only 
on A n + 1 , the second only on d , the third only on ρ , 
and the fourth only on Y n . There is a connection here with 
Selick and Walker's work, and with the Hilton matrix calculus, as 
described by Bokor (1991).


Introduction.
In an earlier paper [5], to which this may be regarded as a sequel, the authors introduced a definition of a Hopf invariant which generalized most (but not all) existing definitions.We recall that definition.
Working in the pointed homotopy category Ᏼ, we consider a cogroup A in Ᏼ with comultiplication µ : A → A ∨ A. We now suppose given a map d : A → X and a map ρ : X → Y 1 ∨ Y 2 .If the projections of ρ onto Y 1 and Y 2 are α 1 and α 2 , respectively, so that α i : X → Y i , then ρ is referred to as a copairing of α 1 and α 2 , which are themselves described as copairable (see [5]).
We have a diagram which does not, in general, commute.However, we may embed (1.1) in the larger diagram where j is the canonical map; and, in (1.2), the bottom square and the composite square both commute.That the composite square commutes follows from the relations jµ = ∆ : Let Y 1 Y 2 be the homotopy fiber of j : Then it follows from what we have said that ρd−(α 1 d∨α 2 d)µ lifts to Y 1 Y 2 .Moreover, analysis of j : Y 1 ∨Y 2 → Y 1 ×Y 2 shows that the lift is unique.We call this lift, as an element of [A, Y 1 Y 2 ], the Hopf invariant of d with respect to ρ, and write it as HI ρ (d).
In this paper, we pursue the study of HI ρ , but allow ourselves one further freedom.Instead of starting with a copairing ρ, which may be thought of as a kind of a counion of two maps, we consider a counion of n maps, which we call an n-counion map ρ : The rest of the definition of HI ρ (d) will be essentially the same, except that, for simplicity, we confine our attention to the case in which A is a commutative cogroup.Since our principal interest in defining Hopf invariants in Ᏼ would be in the case in which A is, at least, a double suspension, we argue that this gain in simplicity is obtained at relatively low cost.
With [A, Y ] commutative-indeed, we write it additively, as is customary with the higher homotopy groups-we may bring to bear the notion of orthogonal idempotents on [A, Y ].Thus let π i : Y → Y i , ι i : Y i → Y be the canonical projection onto the ith summand in Y and the corresponding injection.Then π i ι i is the identity on Y i and e i = ι i π i : Y → Y is an idempotent map.Moreover, e i e j = 0, i≠ j. (1.3) Then e i induces an idempotent endomorphism of [A, Y ] which we also call e i ; and if i ≠ j, then e i and e j are orthogonal idempotent endomorphisms.
On the other hand, it is not true that e We call this idempotent e 0 .Then e 0 e j = e j − n i=1 e i e j = 0 and e j e 0 = 0, as shown above.Thus e 0 ,e 1 ,...,e n form a complete Pierce-like system of orthogonal idempotents on [A, Y ], and we may say that it is the nontriviality of e 0 which allows us, or requires us, to define a Hopf invariant.Notice that the commutativity of [A, Y ] has greatly facilitated this last discussion.
In Section 2, we define the Hopf invariant of d : A → X with respect to the n-counion map ρ : X → Y as the lift of e 0 (ρd) to the homotopy fiber of the canonical map j : We show in what sense the Hopf invariant-which is a homomorphism of commutative groups-is natural and we analyse it as a sum of 2 n − n − 1 elements, each factoring through a given space determined by a summand that such an analysis is vacuous in the case n = 2.We also relate our Hopf invariant to the one given in [4] in connection with the calculation of relative attaching maps for Thom spaces.
In Section 3, we make an entirely different analysis of the Hopf invariant, representing it as the composition of four maps, each depending on some aspect of the original data.One of the constituent maps appears to be closely related to a very general kind of Hopf invariant defined by Walker [11].
The constructions and arguments in this paper are all carried out in the pointed homotopy category Ᏼ.However, they may be couched in categorytheoretic language and executed, with minor modifications, in a general category possessing pullbacks, coproducts, and zero object (see [5]).In particular, we might, as in [5], study these ideas in the category of groups.We might also consider the dual concepts.Indeed, the dual concept is related to a paper of Selick [10].
In Section 4, we show the relationship of the idempotents of this paper, of span 2, to the matrix calculus introduced in [7] and exploited in [2].
Since we always work on Ᏼ, we do not regard it as always necessary to mention the base point explicitly nor to insist on distinguishing notationally between a map and its homotopy class.

The Hopf invariant and naturality.
Let the space Y be given as a wedge (coproduct) (2.1) We may describe Y as a costructured space, with summands Y i .Thus we associate with Y the projections and injections ..,n.We say that the idempotents e i are associated with the costructure on Y .Notice that these idempotents are orthogonal, that is, e i e j = 0, i ≠ j.
Let A be a commutative cogroup in Ᏼ (e.g., a double suspension).Then [A, Y ] is a commutative group and the idempotents e i induce an orthogonal system of idempotent endomorphisms, which we also denote by e i , (2.4) In general, the system {e i } of idempotents on [A, Y ] is not complete; for example, if A = S 3 and Y = S 2 ∨ S 2 , then there is a Whitehead product element [Id 1 , Id 2 ] ∈ π 3 (S 2 ∨ S 2 ) of infinite order, not expressible as e 1 α 1 + e 2 α 2 .

However, for any f ∈ [A, Y ], we set
(2.5) Proposition 2.1.The function f f 0 is an idempotent endomorphism e 0 of [A, Y ].Moreover, the system {e 0 ,e 1 ,...,e n } is a complete system of orthogonal idempotents on [A, Y ].
Proof.Write f 0 = e 0 f .Then e 0 (f . Thus e 0 is an endomorphism.Also e j e i f = e j f − e j f = 0, j= 1, 2,...,n. (2.6) Thus, e 0 f 0 = f 0 , so e 0 is idempotent; and, by the argument above, e j e 0 = 0, j = 1, 2,...,n.The formal calculation would indeed have sufficed, and it shows equally well that e 0 e j = 0. We conclude that {e 0 ,e 1 ,...,e n } constitutes a system of orthogonal idempotents on [A, Y ] such that e 0 + e 1 +•••+e n = 1, as was to be proved.Now let j be the natural inclusion and let (Y ) be the homotopy fiber of j.For any f ∈ [A, Y ], consider the diagram (2.9) Hence jf 0 = 0 and f 0 lifts into (Y ).
Lemma 2.2.The lift of f 0 into (Y ) is unique.
Proof.We have the fiber sequence inducing the exact sequence (2.13) is a split short exact sequence.Notice that the validity of this remark depends on the commutativity of A; so does much of the preceding reasoning.We write (f ) for the lift of f 0 into (Y ).Since e 0 and k * are homomorphisms, it follows that We call Y a summand or simply a subspace of the costructured space Y of span k and write |Y | = k.We plainly have maps

16) and inducing maps
with e Y an idempotent map.
In particular, suppose that |Y | = 2.Then, as Y ranges over the subspaces of Y of span 2, we obtain a system of orthogonal idempotents e Y on Y .We also obtain a map with components π Y .Thus (compare the definition of (f ) above) for any map : A → Y , we may lift e Y (2.20) to the homotopy fiber of π (2.19), which we call 2 Y .Moreover, the lift is unique so that where (2) is the lift of − |Y |=2 e Y so that is a fiber sequence.Note that e Y factors through Y .
We now continue the process of analysing .We next have to analyse (2) .We now allow Y to range over all subspaces of Y of span 3. We thus obtain maps with e Y an idempotent map.As before, we obtain a map with components π Y .The maps e Y on 2 Y constitute a system of orthogonal idempotents so that the map (2) : A → 2 Y may be written as e Y (2) + k (3) where (3) is the lift of (2) − |Y |=3 e Y (2) to 3 Y , the homotopy fiber of π of (2.25).
The process terminates when we arrive at the set of subspaces of span (n− 1).We will then write as a sum of 2 n − n − 1 maps, corresponding to the subspaces of Y of span greater that or equal to 2 (including Y itself).Thus, we have proved the following proposition.

Proposition 2.4. The set {e
given as a wedge of spaces.As a corollary, any map : A → Y may be uniquely expressed as the sum of Notice that the mystery of the number 2 n −n−1 is dissolved if one observes the corollary or the equivalent statement.We next discuss the naturality of .Let g : A → B be a homomorphism of commutative cogroups and let (2.28) Proof.We first remark that is obviously a functor : Ᏼ n → Ᏼ, so h is defined.Now, to prove the commutativity of (2.28), we first consider the diagram establishing the commutativity of (2.29).Now we have an obvious commutative diagram (used, in fact, to present as a functor) (2.31) so the commutativity of (2.28) follows from (2.29) and (2.31), using the fact that k is a monomorphism.
We may further refine the naturality as follows.Let Y be a summand of Y of span k ≥ 2 and let Y (f ) be the corresponding component of (f ), regarded as a map Y (f ) : commutes.We leave the details to the reader, remarking that we may reexpress this refinement by asserting the naturality of the idempotents and summation described in Proposition 2.4.
We now formally introduce the Hopf invariant.A map ρ : X → Y is called an n-counion map or, more precisely, an n-counion of the maps π i ρ : X → Y i , i = 1, 2,...,n.If n = 2, this is called a copairing (see [5]).Now ρ induces a homomorphism ρ : We say that the Hopf invariant is thereby expressed as a sum of constituent subinvariants.
The naturality of the Hopf invariant now expresses itself as follows.We suppose a given commutative diagram where g is a homomorphism of commutative cogroups and h is costructurepreserving.Then (2.33) induces a commutative diagram (

2.35)
There is also a refined form of this naturality statement, based on (2.32), involving the constituent subinvariants.It is further of interest to interpret the vanishing of the Hopf invariant; thus Theorem 2.9.The vanishing of HI ρ (d) is equivalent to the assertion that ρd = e 1 ρd + e 2 ρd +•••+e n ρd. (2.36) A version of Theorems 2.5, 2.7, 2.8, and 2.9 appeared in [3].The above definition of the Hopf invariant generalizes that given in the case n = 2 in [5], where it is further related to a number of existing definitions.Staying always with the case n = 2, but confining attention to coactions (or cooperations, see [6]) ρ, there is a treatment in [4] which is specialized to a stage in the version of Hilton and Milnor [1,8] when the coaction is a comultiplication.
In the same way, we may consider, for each 1) P 1 ,P 2 ,..., Pj ,...,P n , j = 1, 2,...,n. (2.39) Now the union of the spaces T (1) (P 1 ,P 2 ,..., Pj ,...,P n ) is T (2) (P 1 ,P 2 ,...,P n ) "the next fattest wedge," that is, the subspace of P 1 × P 2 ×•••× P n consisting of points with at least two coordinates at the base point.Moreover, the maps w n−1,j combine to produce a map whose mapping cone is precisely T (1) .In general, given a map f : A → B with mapping cone C f , there is, as explained in [6], a cooperation or coaction ρ : (1)  (2.41) and thus an (n+1)-counion map.The map (2.41) may be fed into our definition to produce essentially the Hopf invariant of [4] and to motivate our definition in this paper.

A canonical factorization of the Hopf invariant.
In this section we describe a canonical expression for HI ρ (d) as a composition of four maps, each depending on a particular ingredient of the definition of the Hopf invariant. Given where A is a commutative cogroup, we may express f 0 as the composition where µ : A → A ∨ A is the comultiplication.Further factorizing the right-hand map in (3.1) and slightly modifying the relevant factors yield the composition of f 0 , where σ is the sum of the identity 1, mapping A to the first summand in n+1 A, and maps −1, mapping A into the second, third, ...,(n+1)th summand of n+1 A. We could write σ Now let j, as before, be the canonical map from i B i to i B i .We may then embed (3.2) in the commutative diagram where y appears in the mth factor Y in n Y .Further, it is clear that ∆,j σ = 0 : Thus σ lifts to κ : A → F A , where F A is the homotopy fiber of ∆,j , and we may embed (3.3) in the commutative diagram and obviously We now revert to the Hopf invariant.We only have to replace f in (3.5) by Thus it is a matter of factoring the (F f , ∨f , f )-column of (3.5) in the obvious way to obtain and the factorization This is the canonical factorization of the title of this section.
If the counion of size n is assumed to be given, then we notice that, in (3.9), (i) κ depends only on A; (ii) F d depends only on d; (iii) F ρ depends only on ρ; (iv) λ depends only on Y , with its costructure.Without going into details, we make the obvious remark that a similar factorization is available for each of the 2 n − n − 1 constituent subinvariants of the Hopf invariant in the sense of Theorem 2.8.

Remarks. (i)
The factor F ρ of the Hopf invariant HI ρ (d) in (3.9) is related to Walker's version of the Hopf invariant [11].Walker starts with a pair of maps and constructs the double mapping cylinder Z = Z f ,g .There is thus a commutative square (3.11) Let P be the homotopy pullback of i B and i C in (3.11), creating a commutative diagram (3.12) We now pinch A to a point in (3.11).We think of B and C in (3.11) as having been replaced by the mapping cylinders of f and g, respectively, so that (3.11) becomes a strict pushout of inclusion maps.When we pinch A to a point throughout (3.11), the mapping cylinders become mapping cones C f and C g and Z becomes the one-point union (coproduct) C f ∨C g .Thus (3.11) (3.13) Following Walker, we designate the homotopy pullback of ι f and ι g by C f * C g .We thus obtain the diagram and the pinching maps induce a map of diagram (3.12) to diagram (3.14).The component of this map of the diagrams is Walker's Hopf invariant.If we write ρ for the component of this map of diagrams, then this ρ is a two-counion map and plays the same role as in our definition of the Hopf invariant.
We conjecture that Walker's Hopf invariant is closely related to the map F ρ of (3.8), with X = Z and Y = C f ∨ C g .
(ii) It is plain that the maps κ of the factorization (3.9) for various values of n may be related.We write κ n and σ n for the maps κ and σ of (3.8).Then σ 1 = 1 1 − 1 2 : A → A ∨ A and there are maps where r projects off the second summand A. Then Now let q : n A → n+1 A map a to ( * ,a) and let r : commutes so that there are induced maps, which we again write as q, r , thus We may use the maps q to pass to the limit F A∞ , obtaining a map κ ∞ : A → F A∞ which is independent of n and thus truly universal.
(iii) There is a relation between the duals of the constructions in this paper and some aspects of Selick's paper [10].In [10, Section 3, page 408], Selick's D is a dual for the present Hopf invariant based on two summands for the particular case of a multiplication.Selick's D n is a sum of duals of the Hopf invariant associated to e Y , |Y | ≥ 2, in the case of a multiplication.Selick's f i is the dual of our e Y f for the case |Y | = i and multiplication.In [10, Lemma 1, page 408], he proves that the n-fold multiplication of f is a sum of 2 n − 1 summands.In the proof of [10, Lemma 5, page 410], he starts with the summands of [10, Lemma 1] and then moves the n summands that correspond to e Y , |Y | = 1, to the side of f , getting on the right-hand side a sum of 2 n − n − 1 summands that correspond to e Y , |Y | ≥ 2, mirroring our result in Theorem 2.8.

The Hilton quadratic matrix calculus.
The quadratic matrix calculus defined in [7] and used in [2,7,9] can be obtained by using the present paper idempotents of span 2. We here use [2] as a source and deduce some of the results of [2, Section 4].
Let α : It can be observed that for |Y | = 2, Y is the fiber of the map S 2m ∨ S 2m → S 2m ×S 2m , which, as in [4], is equivalent to Ω(S 2m ) * Ω(S 2m ), which, by James' celebrated formula [8], equals Also, for |Y | = 3, say Y = {a, b, c}, Y can be embedded in the following commutative diagram: and in particular where f i,j factors as Thus f i,j is determined completely by the degree of the map S 4m−1 → S 4m−1 , denoted by a ij on [2, page 373].
For i = j, f i,j = f j factors as Now π j f : S 4m−1 → S 2m has a Hilton-Milnor invariant obtained by considering ρf : S 4m−1 → S 2m ∨S 2m , where ρ is the comultiplication map.The invariant of f is a map a i,i :S 4m−1 → S 4m−1 .Thus the following matrix is obtained: defined on [2, page 373] and called the Hilton-Hopf quadratic form of f .Given any map f : J S 4m−1 → K S 2m , it is determined by the inclusions , each of which is determined by a K × K integer matrix as above.Then f is determined by J integer matrices which can be written using juxtaposition as a single K × JK integer matrix composed of J blocks: j , which are determined by degrees.Thus φ = {deg(φ j,i )} = A(φ) is a K × K matrix, defined exactly as in [2,Lemma 4.5(iii)].(The reader may find, in what follows, some superficial changes of notation from that of [2].)We now describe the matrix calculus.
Suppose the following given composition: Then ψ is determined by a J × J integer degree matrix A(ψ), φ by a K × K integer degree matrix A(φ), and f leads to an integer K × JK matrix of the Hilton quadratic form H(f ).Also φf ψ creates an integer K × JK matrix of the quadratic form H(φf ψ).Then, for i < j, the (i, j)th term of the matrix H(φf ψ) is given by using the projection π i,j φf ψ : Map (4.8) can be presented using inclusions in the following way: for 1 ≤ i < j ≤ K and 1 ≤ k ≤ J. Thus there are K × JK terms as above.Then we have the composition (4.10)This last map equals

.11)
As e s factors through S 2m and is null into S 4m−1 , then com k i,j equals 1≤p<q≤K, 1≤r ≤J π i,j φe p,q f e r ψ inc k .(4.12) Any of the summands is a composition of three maps: which is, by definition, H(f ) r p,q ; and the map of the fibers S 4m−1 p,q → S 4m−1 i,j obtained from (4.14) is induced by applying the matrix φ p,q i,j , which is a 2 × 2 submatrix of A(φ).
Thus we get the formula . Now we consider the matrices as written in [2].
Thus H(f ) r p,q is written in the (p, (r −1)K +q)th spot.While H(f ) k i,j is written in the (i, (k − 1)K + j)th spot.The way to express the indices of matrices so as to accord with the multiplication is (i, p) p, (r − 1)K + q (r − 1)K + q, (k − 1)K + j . (4.17) The correct matrix setup for this is A(φ)H(f )(A(ψ) ⊗ A(α) t ), which is the first line of [2,Lemma 4.8].

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

Theorem 2 . 5 .
The set {ke Y | Y ⊂ Y ,|Y | ≥ 1} is a complete system of 2 n − 1 orthogonal idempotents on the group [A, Y ], obtained from the fact that Y is a wedge of n subspaces.Thus any map f : A → Y may be uniquely expressed as the sum of 2 n − 1 terms f Y , each factoring through a subspace k−1 Y of Y that corresponds to a subspace Y of Y of span k ≥ 1. Remark 2.6.In Theorem 2.5, the idempotent ke Y simply means e Y if |Y | = 1.

Theorem 2 . 8 .
[A, X] → [A, Y ] and, for d : A → X, we define the Hopf invariant of d, relative to ρ, to be (ρd) ∈ [A, Y ].Plainly this defines a homomorphism HI = HI ρ : [A, X] → [A, Y ]. (2.33)We may now invoke Proposition 2.4, applied to the map = (ρd) = HI ρ (d), to conclude The Hopf invariant HI ρ : [A, X] → [A, Y ] may be expressed as the sum of 2 n − n − 1 homomorphisms HI Y , each factoring through a group [A, k−1 Y ] corresponding to a subspace Y of Y of span k ≥ 2.

1 b) ∧ S 2m− 1 c
well-known Hilton-Milnor Theorem, the first cell in Y a,b,c is(S 2m−1 a * S 2m−which is (6m − 3)-connected so that [A, Y a,b,c ] = 0. Thus the only idempotents in Theorem 2.5 correspond to span less than three.In particular, on [A, Y ],

7 )
as described on [2, page 375].Finally, for a cogroup A and Y = K S m , any map φ : A → Y has a decomposition as above: φ = |Y |≥1 e Y φ.If A = Y = K S m , then every e Y , |Y | > 1, factors through a multiconnected space and is thus null.Thus we have φ = K j=1 φ j , where φ j is a map Y → Y π j → S m j Y .Each map out of a wedge is determined by the restrictions S m j Y φ j → S m j so that φ : .15) The map in (4.13) is by definition A(ψ) k r .The second part (4.14) defines a lifting S 4m−1 r → S 4m−1 p,q

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation