Read e-book online Non-Euclidean Geometry PDF

By Henry Parker Manning

ISBN-10: 1418179116

ISBN-13: 9781418179113

ISBN-10: 1603861432

ISBN-13: 9781603861434

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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.

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Non-Euclidean Geometry by Henry Parker Manning


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