Download PDF by Hopf H.: A New Proof of the Lefschetz Formula on Invariant Points

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Extra info for A New Proof of the Lefschetz Formula on Invariant Points

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These axioms are obviously satisfied. MC3. Weak equivalences and fibrations are clearly closed under retracts. Since the construction MΨ is natural with respect to Ψ, cofibrations are also preserved by retracts. MC4. Let the following be a commutative square in F unb (K, M): F Ψ  G GE Φ  GB where Ψ and Φ are respectively either a cofibration and acyclic fibration, or an acyclic cofibration and fibration. We need to show that in the above diagram there exists a lift Ω : G → E. We are going to construct Ωσ : G(σ) → E(σ) by induction on the dimension of σ.

We need ∼ ∼ to show that Ψ can be expressed as composites F → F G and F → G G. As in the previous case, to construct diagrams F : K → M, G : K → M and appropriate natural transformations we argue by induction on the dimension of simplices in K. 34 II. HOMOTOPY THEORY OF DIAGRAMS Let dim(σ) = 0. We define F (σ) and G (σ) to be any objects that fit into the following factorizations of Ψσ in M:  ∼ G  G G G(σ) G G (σ) ∼ G G G(σ) F (σ)  F (σ)  F (σ) @A BCy @A BCy Ψσ Ψσ Let us assume that we have constructed F (σ), G (σ), and appropriate maps for all the simplices σ ∈ K whose dimension is less than n.

For all 0-dimensional simplices σ we take cofibrant replacements QF (σ) F (σ) and sum them up: QF (σ). We then go on by attaching generalized cells along their boundaries. Let τ : ∆[n] → K be a non-degenerate simplex. Assume that we already know how to construct the ocolimit of F on a subcomplex N → K containing the boundary of τ . We then turn the morphism colim∂∆[n] QF → F (τ ) ∼ into a cofibration colim∂∆[n] QF → QF (τ ) F (τ ) and glue a generalized cell QF (τ ) to ocolimN F along its boundary colim∂∆[n] QF ; we take the push-out: ocolim(N∪∂∆[n] ∆[n]) F = colim ocolimN F ← colim∂∆[n] QF → QF (τ ) .

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A New Proof of the Lefschetz Formula on Invariant Points by Hopf H.


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