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By Hassi S., Sebestyen Z., Snoo H.

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6) Lemma. The composite O× k l = l(K, U ): H(K) −−−− −→ H(U, U \ K) ⊗ H(K) ∆ ⊗id ∗ −−− −−→ H(U, U \ K) ⊗ H(U ) ⊗ H(K) id⊗t ∗ −−−−− → H(U, U \ K) ⊗ H(K) ⊗ H(U ) d ⊗id × ∗ −−− −−→ Q ⊗ H(U ) −→ H(U ) coincides with the linear map i∗ : H(K) → H(U ). 7) Remark. Dold’s Lemma was given in terms of singular homology in [Do-M] (cf. also [Do2]). 1), clearly follows from the original statement of Dold’s Lemma. We recall the Alexander duality theorem (cf. 8) Theorem. If A is a compact subset of the Euclidean space Rn+1 then for every k ≥ 0 the vector spaces H 0n−k (Rn \ A) and H k (A) are linearly isomorphic, where H 0n−k denotes the (n − k)-reduced singular homology functor and H k (as in ˇ Section 6) is the k-th Cech cohomology functor.

Then we have f∗q = lim {fα∗q } for every q. −→ From the functoriality of lim we deduce that H: E → A is a covariant functor. −→ ˇ The functor H is said to be the Cech homology functor with compact carriers. We note that if (X, X0 ) is a compact pair, then the family consisting of the single pair (X, X0 ) is a cofinal subset of M = {(Aα , A0α )} for (X, X0 ), and hence we obtain H∗(X, X0 ) = H(X, X0 ). Similarly, if f: (X, X0 ) → (Y, Y0 ) is a map of compact pairs, then H∗(f) = H(f). The following properties of H clearly follow from the Eilenberg–Steenrod axioms for H∗ and some simple properties of lim .

Denote by H(X) the reduced homology vector space of X (see [ES-M] or [Sp-M]). 12) Proposition. Let A be a closed acyclic subset of X. Then for every covering α ∈ Cov X there exists a refinement β ∈ Cov X of α such that the homomorphism iβα∗ : H∗(N (β)|St2 (A,β) ) → H∗(N (α)|St(A,α) ) is a trivial homomorphism of vector spicas. Proof. We recall that the coefficients are in a field F . Hence H∗ (N (α)) are finite-dimensional graded vector spaces. 10) we can find a covering γ ∈ Cov X such that the homomorphism iγα∗ : H∗(N (γ)|St(A,γ) ) → H∗ (N (α)|St(A,α)) is trivial.

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A canonical decomposition for linear operators and linear relations by Hassi S., Sebestyen Z., Snoo H.


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