Axioms for a local Reidemeister trace in fixed point ...

Axioms for a local Reidemeister trace in fixed point and coincidence theory P. Christopher Staecker http://www.messiah.edu/∼cstaecker July 2007

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The Reidemeister trace f → X, f choose a Given f : X → X and fe : X f and we have Zπ-basis for Cq (X),

T =

X

(−1)q tr(feq ) ∈ Zπ

q

The projection into the Reidemeister (twisted conjugacy) classes gives the Reidemeister trace: RT (f, fe) = ρ(T ) ∈ ZR(f ) RT does not depend on the choice of Zπ basis or the simplicial structure, but does depend (in a predictable way) on the choice of lift fe. Wecken’s Trace Theorem: RT (f, fe) =

X

ind([α])[α],

[α]∈R(f )

where ind([α]) = ind(f, p(Fix(αfe))) 2

Properties of the Reidemeister trace

• (Homotopy) If f ' f 0 and this homotopy lifts to fe ' fe0, then RT (f, fe) = RT (f 0, fe0).

• (Lift invariance) If RT (f, fe) =

X

k[σ][σ],

[σ]

then RT (f, αfe) =

X

k[σ][σα−1]

[σ]

• (Fixed point of lift) If [α] appears in RT (f, fe), then αfe has a fixed point.

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Roadmap We will use the above (sometimes weakened) as axioms, and obtain a uniqueness theorem for the Reidmeister trace in coincidence theory using a similar result (S. 2007) for the local coincidence index.

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The uniqueness of the local coincidence index Theorem 1 (S. 2007). For maps f, g : X → Y of orientable differentiable manifolds of the same dimension with U ⊂ X open with compact closure, there is a unique integer-valued function ind(f, g, U ) satisfying: • (Normalization) ind(f, g, X) = L(f, g). • (Homotopy) If f, g are admissably homotopic to f 0, g 0, then ind(f, g, U ) = ind(f 0, g 0, U ). • (Additivity) If Coin(f, g, U ) ⊂ A t B, then ind(f, g, U ) = ind(f, g, A) + ind(f, g, B). In the case X = Y , orientability is not needed, and the normalization axiom can be weakened to: ind(const, id, X) = 1. 5

A local Reidemeister trace Fares and Hart [1994] gave a local Reidemeister trace RT (f, fe, eı, U ) ∈ ZR(f ), and showed that it satisfies:

• (Homotopy) If f ' f 0, and this homotopy lifts to fe ' fe0, then RT (f, fe, eı, U ) = RT (f 0, fe0, eı, U ). • (Additivity) If Fix(f, U ) ⊂ A t B, then RT (f, fe, eı, U ) = RT (f, fe, eı, A)+RT (f, fe, eı, B). P e e • (Lift Invariance) If RT (f, f , ı, U ) = [σ] iσ [σ],

then RT (f, αfe, eı, U ) =

X

iσ [σα−1]

[σ] 6

Our uniqueness result We use the above as axioms to show that the local Reidemeister trace is unique in coincidence theory (and fixed point theory). Letting π = π1(Y ), the Reidemeister classes R(f, g) are the classes with respect to doubly twisted conjugacy: [α] = [β] ⇐⇒ α = ψ(σ)−1βϕ(σ) where ϕ = π1(f ) and ψ = π1(g). Our theorem is: Theorem 2. If X, Y are orientable differentiable manifolds of the same dimension, there is a unique function RT(f, fe, g, ge, U ) ∈ ZR(f, g) satisfying five axioms: 7

• (Normalization) c(RT (f, fe, g, ge, X)) = L(f, g) • (Additivity) If Coin(f, g, U ) ⊂ A t B, then RT (f, fe,g, ge, U ) = RT (f, fe, g, ge, A) + RT (f, fe, g, ge, B). • (Homotopy) If f ' f 0 and g ' g 0 and these homotopies lift to fe ' fe0 and ge ' ge0, then RT (f, fe, g, ge, U ) = RT (f 0, fe0, g 0, ge0, U ). • (Lift invariance) c(RT (f, fe, g, ge, U )) = c(RT (f, αfe, g, β ge, U )). • (Coincidence of lifts) If [α] appears in RT (f, fe, g, ge, U ), then αfe and ge have a coe. incidence on U 8

Connections to the index Lemma 3. If c : ZR(f, g) → Z is the sum of the coefficients, then c(RT(f, fe, g, ge, U )) = ind(f, g, U ). Proof. By the lift invariance axiom, c◦RT does not depend on fe or ge. Consider c ◦ RT (f, g, U ) ∈ Z The function c ◦ RT is additive, homotopy invariant, and gives the Lefschetz number if U = X. These are the three axioms for the coincicdence index, and so c ◦ RT is the coincidence index. (Note: didn’t need the Coincidence of Lifts axiom.)

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Coincidence classes Lemma 4. Given a tuple (f, fe, g, ge, U ), there is a homotopic tuple having isolated coincidence points (on the same U ). Thus we can assume without loss of generality (by the homotopy axiom) that our maps have isolated coincidence points. We also know that these coincidence points are partitioned into coincidence classes of the form e ) ⊂ X. C[α] = pX Coin(αfe, ge, U

If [α] = [β], then C[α] = C[β]. If [α] 6= [β], then C[α] and C[β] are disjoint.

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For a coincidence point x, let [x] be the Reidmeister class [α] for which x ∈ C[α]. Theorem 5. For each coincidence point x ∈ U , let Ux be an isolating neighborhood of x. Then X

RT(f, fe, g, ge, U ) =

ind(f, g, Ux)[x]

x

Proof. Clearly RT (f, fe, g, ge, U ) =

X x

RT (f, fe, g, ge, Ux)

by the additivity axiom. It remains to show that RT (f, fe, g, ge, Ux) = ind(f, g, Ux)[x].

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It remains to show that RT (f, fe, g, ge, Ux) = ind(f, g, Ux)[x]. First observe that RT (f, fe, g, ge, Ux) is an integral multiple of [x]: If RT contains a term [β], then by the coincidence of lifts axiom β fe and ge have a coincidence above Ux. But since x is the only coincidence point in Ux, it must belong to the coincidence class C[β], which is to say that [β] = [x]. Thus we know that RT (f, fe, g, ge, Ux) = k[x], and we need only show that k = ind(f, g, Ux). We already know that the sum of coefficients in RT equals the index, so we are done.

Uniqueness The above gives a formula for the computation of RT , which means that it must be unique (and also that it exists). We also have shown a coincidence version of Wecken’s trace theorem: RT (f, fe, g, ge, U ) =

X

ind(C[α])[α].

[α]

This leads fairly easily to a strong lift invariance property: If RT (f, fe, g, ge, U ) =

X

k[σ][σ],

[σ]

then RT (f, αfe, g, β ge, U ) =

X

k[σ][βσα−1].

[σ]

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Selfmaps In the case X = Y , we obtain a uniqueness result without requiring orientability, using only a weak normalization axiom: If f is a constant map, then f U) = 1 c ◦ RT (f, fe, id, id,

This result specializes directly to the fixedpoint local Reidemeister trace of Fares and Hart.

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A lingering question Are all five axioms needed to establish uniqueness? Just the first four do not suffice: defining RT (f, fe, g, ge, U ) = ind(f, g, U )[1] satisfies normalization, homotopy, additivity, and lift invariance, but is not the “correct” Reidemeister trace. But is the lift invariance axiom necessary?

?

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