Geometric and homotopy theoretic methods in Nielsen coincidence theory

In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. Here we extend it to pairs (f_1, f_2) of maps between manifolds of arbitrary dimensions. This leads to estimates of the minimum numbers MCC(f_1, f_2) (and MC(f_1, f_2), resp.) of pathcomponents (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to (f_1, f_2). Furthermore we deduce finiteness conditions for MC(f_1, f_2). As an application we compute both minimum numbers explicitly in four concrete geometric sample situations. The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space E(f_1, f_2) into pathcomponents. Its higher dimensional topology captures further crucial geometric coincidence data. An analoguous approach can be used to define also Nielsen numbers of certain link maps.


Introduction and discussion of results
Throughout this paper f 1 , f 2 : M → N denote two (continuous) maps between the smooth connected manifolds M and N without boundary, of strictly positive dimensions m and n, resp., M being compact.
We would like to measure how small (or simple in some sense) the coincidence locus  [BGZ], 1.1). It coincides with the minimum number min{#C(f ′ 1 , f 2 )|f ′ 1 ∼ f 1 } where only f 1 is modified by a homotopy (cf. [Br]). In particular, in topological fixed point theory (where M = N and f 2 is the identity map) this minimum number is the principal object of study (cf. [B], p. 9).
In higher codimensions, however, the coincidence locus is generically a manifold of dimension m − n > 0, and M C(f 1 , f 2 ) is often infinite (see e.g. examples I and III below). Thus it seems more meaningful to study the minimum number of coincidence components ) denotes the (generically finite) number of pathcomponents of the indicated coincidence subspace of M .

Typeset by A M S-T E X
Question. How big are M CC(f 1 , f 2 ) and M C(f 1 , f 2 )? In particular, when do these invariants vanish, i.e. when can the maps f 1 and f 2 be deformed away from one another?
In this paper we discuss lower bounds for M CC(f 1 , f 2 ) and geometric obstructions to M C(f 1 , f 2 ) being trivial or finite.
Remark 1.10. In various situations some of the estimates spelled out in part (iii) of this theorem are known to be sharp (compare also [K 3]). E.g. in the selfcoincidence setting (where f 1 = f 2 ) we have always M CC(f 1 , f 2 ) ≤ 1 (since here C(f 1 , f 2 ) = M ). In the "root setting" (where f 2 maps to a constant value * ∈ N ) all Nielsen classes are simultaneously essential or inessential (since our ω-invariants are always compatible with homotopies of (f 1 , f 2 ) and hence, in this particular case, with the action of π 1 (N, * ), cf. the discussion in [K 3] following 1.11). Therefore in both settings M CC(f 1 , f 2 ) is equal to the Nielsen number N (f 1 , f 2 ) provided ω(f 1 , f 2 ) = 0 (and n = 2 if f 2 ≡ * ).
Further geometric and homotopy theoretic considerations allow us to determine the Nielsen and minimum numbers explicitly in several concrete sample situations (for proofs see § 6 below).
Example I. Given integers q > 1 and r, let N = CP (q) be q-dimensional complex projective space, let M = S(⊗ r C λ C ) be the total space of the unit circle bundle of the r th tensor power of the canonical complex line bundle, and let f : M → N denote the fiber projection. Then and As was shown above (cf. 1.10), in any selfcoincidence situation (where f 1 = f 2 ) M CC(f 1 , f 2 ) must be 0 or 1 and it remains only to decide which value occurs. In the previous example this can be settled by the normal bordism class ω(f, f ) ∈ Ω 1 (M ; ϕ), a weak form of ω(f, f ) which, however, captures a delicate ("second order") Z 2 -aspect as well as the dual of the classical first order obstruction. Already in this simple case standard methods of singular (co)homology theory yield only a necessary condition for M CC(f 1 , f 2 ) to vanish (cf. [DG], 2.2). In higher codimensions m − n the advantage of the normal bordism approach can be truely dramatic.
Example II. Given natural numbers k < r, let M = V r,k (and N = G r,k, , resp.) be the Stiefel manifold of orthonormal k-frames (and the Grassmannian of k-planes, resp.) in R r . Let f : M → N map a frame to the plane it spans.
Assume r ≥ 2k ≥ 2. Then Here the normal bordism obstruction ω(f, f ) ∈ Ω m−n (M ; ϕ) (cf. 4.7) contains precisely as much information as its "highest order component" where [SO(k)] denotes the framed bordism class of the Lie group SO(k), equipped with a left invariant parallelization; the Euler number χ(G r,k ) is easily calculated: it vanishes if k ≡ r ≡ 0(2) and equals [r/2] [k/2] otherwise. Without loosing its geometric flavor, our original question translates here -via the Pontryagin-Thom isomorphism -into deep problems of homotopy theory (compare the discussion in the introduction of [K 2]). Fortunately powerful methods are available in homotopy theory which imply e.g. that M CC(f, f ) = M C(f, f ) = 0 if k is even or k = 7 or 9 or χ(G r,k ) ≡ 0(12); however, if k = 1 and r ≡ 1(2), or if k = 3 and r ≡ 1(12) is odd, or if k = 5 and r ≡ 5(6), then M CC(f, f ) = M C(f, f ) = 1. These results seem to be entirely out of the reach of the methods of singular (co)homology theory since we would have to deal here with obstructions of order m − n + 1 = k(k−1) 2 + 1.
Example III. Let N be the torus (S 1 ) n and let ι 1 , . . . , ι n denote the canonical generators of H 1 ((S 1 ) n ; Z). If the homomorphism has an infinite cokernel (or, equivalently, the rank of its image is strictly smaller than n), then On the other hand, if the cup product is nontrivial then M C(f 1 , f 2 ) = ∞ whenever m > n; if in addition n = 2, then M CC(f 1 , f 2 ) equals the (finite) cardinality of the cokernel of f 1 * − f 2 * (cf. 1.12).
In the special case when N is the unit circle S 1 we have: An important special case of our invariants are the degrees of a given map f : M → N (here * denotes a constant map).
Example IV (homotopy groups). Let M be the sphere S m ; in view of the previous example we may also assume that n ≥ 2.
and Ω m−n (M ; ϕ) with the corresponding groups in the top line of the diagram (This is possible since the loop space ΩN occurs as a typical fiber of the natural projection p : E(f 1 , f 2 ) → S m , cf. [K 3], § 7 and [K 5]). Furthermore, after deforming the maps f 1 and f 2 until they are constant on opposite halfspheres in S n , we see that

T T T T T T T T T T T T T T T T
and similarly for ω # and ω.
Thus it suffices to study the degree maps in diagram 1.14. They turn out to be group homomorphisms which commute with the indicated natural forgetful homomorphisms.
Special case: M = S m , N = S n , n ≥ 2. Here deg # is injective and we see that for all maps f : S m → S n . There are many dimension combinations (m, n) where the equality N (f, * ) = N # (f, * ) is also valid for all f or, equivalently, where deg is injective (compare e.g. our remark 4.10 below or [K 3], 1.16). However, if n = 1, 3, 7 is odd and m = 2n − 1, or if e.g. (m, n) = (8, 4), (9, 4), (9, 3), (10, 4), (16,8), (17,8), (10 + n, n) for 3 ≤ n ≤ 11, or (24, 6), then there exists a map f : Very special case: Finally let us point out that our approach can also be applied fruitfully to study linking phenomena. Consider e.g. a link map the closed manifolds M 1 and M 2 have disjoint images). Just as in the case of two disjoint closed curves in R 3 the degree of linking can be measured to some extend by the geometry of the overcrossing locus: it consists of that part of the coincidence locus of the projections to N , where f 1 is bigger than f 2 (w.r. to the R-coordinate). Here the normal bordism/pathspace approach yields strong unlinking obstructions which, in addition, turn out to distinguish a great number of different link homotopy classes. Moreover it leads to a natural notion of Nielsen numbers for link maps (cf. [K 4]).

The path space
A crucial feature of our approach to Nielsen theory is the central role played by the space E(f 1 , f 2 ). It yields the Nielsen decomposition of coincidence sets in a very natural geometric fashion. In the defining equation 1.7 N I denotes the space of all continuous paths θ : to yield the Hurewicz fibration is just the loop space Ω(N, y 0 ) of paths in N starting and ending at y 0 = f 1 (x 0 ) = f 2 (x 0 ); let θ 0 denote the constant path at y 0 .
Proposition 2.3. The sequence of group homomorphisms The proof is fairly evident. In fact, we are dealing here essentially with the long exact homotopy sequence of the fibration p.

Normal bordism
In this section we recall some standard facts about a geometric language which seems well suited to describe relevant coincidence phenomena in arbitrary codimensions.
Let X be a topological space and let ϕ be a virtual real vector bundle over X, i.e. an ordered pair (ϕ + , ϕ − ) of vector bundles written is a stable vector bundle isomorphism (i.e. we can first add trivial vector bundles of suitable dimensions on both sides).
Two such triples (C i , g i , g i ), i = 0, 1, are bordant if there exists a compact singular (q + 1)-dimensional ϕ-manifold (B, b, b) in X with boundary ∂B = C 0 ∐C 1 such that b and b, when restricted to ∂B, coincide with the corresponding data g i and g i at C i , i = 0, 1 (via vector fields pointing into B along C 0 and out of B along C 1 ). The resulting set of bordism classes, with the sum operation given by disjoint unions, is the q th normal bordism group Ω q (X; ϕ) of X with coefficients in ϕ.
Example 3.1. Let G denote the trivial group or the (special) orthogonal group (S)O(q ′ ), q ′ > q+1. For any topological space Y let ϕ + be the classifying bundle over BG, pulled back to X = Y × BG , while ϕ − is trivial. Then Ω q (X; ϕ) is the standard (stably) framed, oriented or unoriented q th bordism group of Y (compare e.g. [CF], I.4 and 8).
For every virtual vector bundle ϕ over a topological space X there are wellknown Hurewicz homomorphisms into singular homology with local integer coefficients Z ϕ (which are twisted like the orientation line bundle ξ ϕ = ξ ϕ + ⊗ ξ ϕ − of ϕ); they map a normal bordism class [C, g, g] to the image of the fundamental class [C] ∈ H q (C; Z T C ) by the induced homomorphism g * .
In most cases µ leads to a big loss of information. However for q ≤ 4 this loss can often be measured so that explicit calculations of (and in) Ω q (X; ϕ) are possible (in particular so when ϕ is highly nontrivial), see theorem 9.3 in [K 1]. We obtain for example Lemma 3.3. Assume X is pathconnected. Then (ii) The following sequence is exact: Here δ 1 (1) is represented by the invariantly parallelized unit circle, together with a constant map, and denote Stiefel-Whitney classes of ϕ.
The setting of (normal) bordism groups provides also a first rate illustration of the fact that the geometric and differential topology of manifolds on one hand, and homotopy theory on the other hand, are often but two sides of the same coin. Indeed, if ϕ − allows a complementary vector bundle ϕ −⊥ (such that ϕ − ⊕ ϕ −⊥ is trivial), then the wellknown Pontryagin-Thom construction allows us to interpret Ω q (X; ϕ), q ∈ Z, as a (stable) homotopy group of the Thom space of ϕ + ⊕ ϕ −⊥ which consists of the total space of ϕ + ⊕ ϕ −⊥ with one point "added at infinity" (compare e.g. [CF], I, 11 and 12). Thus the methods of algebraic topology offer another (and often very powerful) approach to computing normal bordism groups (cf. e.g. chapter II of [CF]).
Example 3.4. The Thom space of the vector bundle ϕ = R k over a onepoint space is the sphere S k = R k ∪ {∞}. Hence the framed bordism group Ω f r q := Ω q ({point}; ϕ) is canonically isomorphic to the stable homotopy group π S q := lim k→∞ π q+k (S k ) of spheres. It is computed and listed e.g. in Toda's tables (in chapter XIV of [T]) whenever q ≤ 19.
For further details and references concerning normal bordism see e.g. [D] or [K 1].

The invariants
In this section we discuss the invariants ω(f 1 , f 2 ) and N (f 1 , f 2 ) based on normal bordism, as well as their sharper (nonstabilized) versions ω # (f 1 , f 2 ) and N # (f 1 , f 2 ). We refer to [K 3] for some of the details and proofs (see also [K 5]).
In the special case when the map (f 1 , f 2 ) : M → N ×N is smooth and transverse to the diagonal (4.1) ∆ = {(y, y) ∈ N × N |y ∈ N } the coincidence set is a smooth submanifold of M . It comes with the maps  where ϕ := f * 1 (T N ) − T M and ϕ := p * (ϕ). Invariants with precisely the same properties can be constructed in general. Indeed, apply the preceding procedure to a smooth map (f ′ 1 , f ′ 2 ) which is transverse to ∆ and approximates (f 1 , f 2 ).
Definition 4.9. A pathcomponent of E(f 1 , f 2 ) is called essential if the corresponding direct summand of ω(f 1 , f 2 ) is nontrivial. The Nielsen coincidence number N (f 1 , f 2 ) is the number of essential pathcomponents A ∈ π 0 (E(f 1 , f 2 )).
Since we assume M to be compact, N (f 1 , f 2 ) is a finite integer. It vanishes if and only if ω(f 1 , f 2 ) does.
Remark 4.10. In the construction above we have neglected an important geometric aspect: C is much more than just an (abstract) singular manifold with an description of its stable normal bundle. If we keep track (i) of the fact that C is a smooth submanifold in M , and (ii) of the nonstabilized isomorphism (4.4), we obtain the sharper invariants ω # (f 1 , f 2 ) and N # (f 1 , f 2 ). Note, however, that the bordism set Ω # (f 1 , f 2 ) in which ω # (f 1 , f 2 ) lies has possibly no group structurethe union of submanifolds may no longer be a submanifold. Also N # (f 1 , f 2 ) = 0 if ω # (f 1 , f 2 ) = 0, but the converse may possibly not hold in general -nulbordisms of individual coincidence components may intersect in M × I.
Example 4.12: the classical dimension setting m = n. Here the coincidence set consists generically of isolated points (in this very special situation the stabilizing map and the Hurewicz homomorphism µ in 4.11 lead to no significant loss of information). In our approach each Nielsen class is expressed as an inverse image of some pathcomponent A of E(f 1 , f 2 ) (compare proposition 2.3). The corresponding index 3) lies in Z 2 precisely if w 1 ( ϕ|A) = 0 or, equivalently, if for some (and hence all) x 0 ∈ g −1 (A) there exists a class α ∈ π 1 (M, x 0 ) such that f 1 * (α) = f 2 * (α) but w 1 (M )(α) = f * 1 (w 1 (N ))(α) (cf. [K 3], 5.2; this agrees with the criterion quoted in [BGZ], p. 53, lines 5-6). If π 1 (N ) is commutative, then either the indices of all Nielsen classes are integers, or they all lie in Z 2 . However, it is easy to construct examples (e.g. involving maps from the Klein bottle to the punctured torus) where both types of pathcomponents A ∈ π 0 (E(f 1 , f 2 )) occur.
In any case our approach makes it clear from the outset where the indices of Nielsen classes must take their values.
In the setting of fixed point theory (where f 2 is the identity map on M = N ) the transition from the ω-to the ω-invariant (cf. 4.6 and 4.7) which forgets the pathspace E(f 1 , f 2 ) parallels the transition from Nielsen to Lefschetz theorywith all the loss of information which this entails.
5. Finiteness conditions for the minimum number MC(f 1 , f 2 ) Consider the following possible conditions concerning the invariants defined in 1.2, 4.6, and 4.7: where direct summation is taken over all A ∈ π 0 (E(f 1 , f 2 )) and i A * is induced by the inclusion of a point z A into the pathcomponent A (and by a local orientation of ϕ at z A ); (C 4) ω(f 1 , f 2 ) lies in the image of a similarly defined homomorphism Proposition 5.1. Each of the first three conditions implies the next one.
Proof. Assume that the coincidence set C(f 1 , f 2 ) is finite. If a generic pair (f ′ 1 , f ′ 2 ) approximates (f 1 , f 2 ) closely enough then each component of C(f ′ 1 , f ′ 2 ) lies in a ball neighbourhood of some x ∈ C(f 1 , f 2 ); moreover the corresponding paths which occur in the construction of ω(f 1 , f 2 ) lie entirely in a ball neighbourhood of y = f 1 (x) = f 2 (x) and hence can be contracted into the constant path at y. Thus (C 2) implies (C 3). The proposition follows.
Our coincidence invariants ω(f 1 , f 2 ) and ω(f 1 , f 2 ) project to the obstructions (5.2) [ ω(f 1 , f 2 )] ∈ coker(i E * ) [ω(f 1 , f 2 )] ∈ coker i * which must vanish if M C(f 1 , f 2 ) is to be finite. For m − n = 0 these cokernels are trivial, M C(f 1 , f 2 ) is actually finite and each integer d ≥ 0 can occur as the value of this minimum number for a suitable pair of maps (e.g. for selfmaps of degrees d and 0 on S 1 ).
Remark 5.3. The finiteness criterion in 5.1 can be sharpened to yield a nonstabilized version involving ω # (f 1 , f 2 ).
For selfcoincidences there is a partial converse of proposition 5.1.
Proof. These conditions are compatible with homotopies of f 1 and f 2 (cf. [K 3], 3.3 and the discussion following 4.4). Hence we may assume that f 1 = f 2 =: f .
Recall that the selfcoincidence invariant ω(f, f ) is just the singularity obstruction ω( R , f * (T N )) to sectioning the vector bundle f * (T N ) over M without zeroes (cf. [K 2], theorem 2.2). Now assume that m < 2n − 2 and ω(f, f ) = i * (ω 0 ) for some ω 0 ∈ Ω f r m−n ∼ = π S m−n . Then there exists a map u ∂ : S m−1 → S n−1 whose (stable) Freudenthal suspension corresponds to ω 0 . Now consider the trivial bundle f * (T N )|B m over some compact ball B m in M and interpret u ∂ as a nowhere zero section over ∂B m = S m−1 .
We will extend u ∂ to a section u of f * (T N ) over all of M which vanishes only in the centre point of B m . Over the ball B m we use the obvious "concentric" extension. Note, however, that generically the zero set of any extension of u ∂ over B m is a framed manifold which represents ω 0 . Thus a generic extension of u ∂ to the complement M − • B m must have a nulbordant manifold of zeroes (representing ω(f, f ) − i * (ω 0 ) = 0). According to theorem 3.7 in [K 1] these zeroes can be removed altogether.
The resulting section u of f * (T N ) allows us to construct a "small" homotopy of f : for every x ∈ M just deform f (x) somewhat in the direction of the tangent vector u(x) ∈ T f (x) (N ). We obtain a map which has only one coincidence point with f .

The examples of the introduction
In view of the selfcoincidence theorem in [K 2] and of our theorem 5.4 the first example is a special case of (ii) the finiteness obstruction µ(ω(f, f )) ≃ [ω(f, f )] (cf. 5.2 and the subsequent discussion in § 5) vanishes if and only if χ(N ) ∈ e(ξ)(H 2 (N ; Z)).
Proof. We will extend the arguments of section 4 in [K 2]. Consider the commuting diagram (6.2) Here the vertical exact sequences are as described in lemma 3.3. The horizontal lines are exact Gysin sequences of ξ in normal bordism and in homology (or, equivalently, oriented bordism), compare [K 1], 9.20 and 9.4. As was shown in [K 2], § 4, we have ω(f, f ) = χ(N ) · ∂(1). Since the Stiefel-Whitney class w 2 (ξ) is the mod 2 reduction of e(ξ), the proposition follows.
Next let us examine example II. If r ≥ 2k ≥ 2 then according to the theorem in the introduction of [K 2] only the "highest order part" 1.11 of the complete obstruction ω(f, f ) (to deforming f away from itself) survives. Thus the finiteness obstruction [ω(f, f )] (cf. 5.2) vanishes. If also k ≥ 2 then it follows from theorem 5.4 that M C(f, f ) (and hence also M CC(f, f )) equals 0 or 1 according to whether ω(f, f ) vanishes or not. It requires using deep results of homotopy theory and of other branches of algebraic topology to decide which of the two values occur actually, but it can be done at least for k ≤ 10 (cf. [K 2], § 3). However, the case k = 1 (where we deal with the standard projection from S r−1 to real projective space) is elementary.
Next we turn to example III where N = (S 1 ) n . Use the Lie group structure to replace (f 1 , f 2 ) by the pair (f, * ) which consists of the quotient f = f 1 · f −1 2 and of the constant map taking values at the unit element of (S 1 ) n . This does not change the coincidence sets and data significantly.
Since each torus (S 1 ) k is a K(Z k , 1)-space the homotopy class of f is determined by f * : H 1 (M ; Z) → H 1 ((S 1 ) n ; Z). Moreover, if the image of f * has rank k < n, then f factors up to homotopy through the lower dimensional torus (S 1 ) k and hence through (S 1 ) n − { * }. On the other hand, if the image of f * has rank n (or, equivalently, the Reidemeister set R(f, * ) ∼ = coker f * is finite) then -according to 1.10 -all Reidemeister classes are essential and hence N (f, * ) = #R(f, * ), , and (f • coll) * , resp., map a generator of H n (S n ; Z) to the cup product ι 1 · · · ι n ∈ H n ((S 1 ) n ; Z), and to the Poincaré dual of µ(ω(f, * )), resp. (compare 4.11 and [GJW], 3.3). Our claims concerning example III in the introduction follow now from § 5.
Finally note that the facts described in example IV follow mainly from the discussion in [K 3] (see 1.14 -1.17 as well as sections 7 and 8); the calculation of Ω f r 1 (ΩS 2 ) can be understood easily with the help of our lemma 3.3. Let us put the role of the pathspace E(f 1 , f 2 ) and its influence on the relative strength of our invariants into perspective (compare diagram 4.11).
In the selfcoincidence situation f 1 = f 2 := f the fibration p : E(f, f ) → M allows a global section s by constant paths; therefore ω(f, f ) = s * (ω(f, f )) is precisely as strong as the (usually much weaker) invariant ω(f, f ) which does not involve any pathspace data. As examples I and II illustrate, ω(f, f ) may nevertheless capture decisive and very delicate information (which is also registered to some extend by the Nielsen number N (f, f ) in spite of the fact that it can take only the values 0 and 1).
In example III our pathspace approach serves to decompose coincidence sets into Nielsen classes. However, it does not seem to enrich the higher dimensional homotopy theoretical aspects of their data very much (as the torus is aspherical; compare 2.3). Still, all natural numbers can occur here as Nielsen numbers of suitable maps f 1 and f 2 .
In contrast, in example IV the higher dimensional topology of E(f 1 , f 2 ) turns out to be potentially very rich (e.g. when N = S n , n ≥ 2) and able to capture much more than just the decomposition into Nielsen classes.