By Henry Parker Manning

ISBN-10: 1418179116

ISBN-13: 9781418179113

ISBN-10: 1603861432

ISBN-13: 9781603861434

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**Extra info for Non-Euclidean Geometry**

**Example text**

Then the matrix of H∗(f): H∗ (X; Q) → H∗(X; Q) has integral coeﬃcients and is equal to a matrix of the induced homomorphism on H∗(X; Z)/Tor (H∗ (X; Z)). Consequently L(f) does not depend on the torsion part of H∗(X, Z). Proof. For given 0 ≤ i ≤ D, let {eil } be a basis of the free part of Hi (X; Z). For the image of eik by Hi(f) we have Hi(f)(eik ) = aikl eil + g l where g is a torsion element and akl ∈ Z. Obviously the matrix of induced map Hi(f): Hi (X; Z)/Tor (Hi (X; Z)) is equal to {ailk }.

Elements corresponding to zy0 by these isomorphisms are denoted zB , zy respectively. In general for a compact subset K ⊂ E we ﬁnd a closed ball B containing K in its interior and we deﬁne the orientation along K as the image of zB (deﬁned above) by the homomorphism ιKB : Hn (E, E \ B) → Hn (E, E \ K) induced by the natural inclusion. 16) Lemma. The deﬁnition of the local orientation along a compact subset K ⊂ E does not depend on the choice of the ball B. It depends only on the choice of the orientation at a point x ∈ K.

Now sfr| (s(x)) = sf(rs(x)) = sf(x) = s(x), hence s(x) ∈ Fix (sfr| ). Let y ∈ Fix (sfr| ). We show that r(y) ∈ Fix (f). Since y ∈ r−1 (U ) and y = sfr(y), r(y) = r(sfr(y)) = fr(y), hence r(y) ∈ Fix (f). The above equality of ﬁxed point sets suggests the following extension of the deﬁnition of the ﬁxed point index. Under the above notation we deﬁne the ﬁxed point index of a compactly ﬁxed map f: U → X, X ∈ ENR, U ⊂ X, open subset, as ind (f) = ind (sfr| ). Since f is compactly ﬁxed, by the above lemma sfr| : r−1(U ) → E is also compactly ﬁxed, hence the index on the right hand side is deﬁned.

### Non-Euclidean Geometry by Henry Parker Manning

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