Differential Geometry

An introduction to differential geometry with use of tensor by Luther Pfahler Eisenhart

By Luther Pfahler Eisenhart

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It will have E2 term given by E2p,q = H p (T n−1 ; Rq ρ∗ Jn ). 1, all of the Rq ρ∗ Jn are zero except R1 . 1, the spectral sequence immediately stabilizes and E∞ = E2 . 14) H p (T n−1 ; Rq ρ∗ Jn ) p+q=m ∼ = H m−1 (T n−1 ; R1 ρ∗ Jn ) where the last line is true because all Rq are zero except R1 . 1, R 1 ρ∗ J n ∼ = Jn−1 . Thus H m (T n ; Jn ) ∼ = H m−1 (T n−1 ; Jn−1 ). The induction is complete. / Z. Then Now, suppose that t ∈ / Zn . 14) are zero. 3. The sheaf Rq π∗ Jn over Rn is 0 for q = n. When q = n, it is a skyscraper sheaf supported on Zn , with each tower isomorphic to C.

Proof. 2 from [Iv]. 1. 4) H m (M, S) = H p (B, Rq π∗ S). p+q=m We will use the Leray spectral sequence twice in the course of this calculation: once with the map (R × S 1 )n → Rn , and once with (S 1 )n → (S 1 )n−1 . Notation. We generally use J to denote the sheaf of sections flat along the leaves. In what follows, use Jn to denote this sheaf over (R × S 1 )n , when we need to be specific about the dimension. We will often need to consider one component of (R × S 1 )n at a time, with other components fixed.

We can use this to define gE on the rest of IO . 4c), we get a similar result for gF ; the case for gG is similar. 8c) (1 − e−2πis )gG = − e−2πis hEF + e−2πis hF G + hGE These extensions define β on the rest of the sets E, F , and G. Finally, there are additional conditions that β must satisfy, which arise from the intersections around the outside of the ‘ring,’ namely EF , F G, and GE. 10) hGE = e−2πis gE − gG on IAO . 8), satisfy these equations with no further restrictions (still assuming e2πis = 1).

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