| R.Bott, L.Tu: Differential Forms in Algebraic Topology, Springer-Verlag, 1982. |
....consequence of formula (3.6) is that the coefficients of M k;n stabilize as n 1. In particular, if we let n 1 in (3.6) we deduce M k 1;1 = M k;1 (t) t M k 1;1 M k 1;1 = 1 1 Gamma t M k;1 (3. 7) which yields the known result about the cohomology of the classifying space of U(k) [BT]) P t (BU(k) 1 (1 Gamma t ) Delta Delta Delta (1 Gamma t 2k (3.8) The formulae (3.6) can also be used to rederive the Poincar e polynomials of the complex grassmanians in the form presented e.g. in [BT] We would also like to mention that the Schubert cells which describe the ring ....
....yields the known result about the cohomology of the classifying space of U(k) BT] P t (BU(k) 1 (1 Gamma t ) Delta Delta Delta (1 Gamma t 2k (3.8) The formulae (3. 6) can also be used to rederive the Poincar e polynomials of the complex grassmanians in the form presented e.g. in [BT]. We would also like to mention that the Schubert cells which describe the ring structure of the cohomology of G k;n can be given a Morse theoretic description. They are precisely the unstable manifolds of our Morse function f . We conclude this section with a different description of our Morse ....
R. Bott, L.Tu: Differential forms in algebraic topology , Springer-Verlag ,1982
.... C p Gamma1;q (M ) the horizontal differential of degree Gamma1) ffi : C p;q (M ) Gamma C p;q Gamma1 (M ) the vertical differential of degree Gamma1) The periodic double complex is defined by (C per ; M ) d; ffi ) C per p;q (M ) q Gammap (M ) p; q 2 Z) Thus, see [5, 16]) there are two spectral sequences associated with this periodic double complex which will be studied in the forthcoming sections. 12. The second spectral sequence Let 0 ffi r be the differential of bidegree (r Gamma 1; Gammar) so that the groups 0 E r 1 p;q (M ) are isomorphic to the ....
R. Bott, L. W. Tu: Differential Forms in Algebraic Topology, GTM 82, Springer-Verlag, Berlin, 1982.
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R.Bott, L.Tu: Differential Forms in Algebraic Topology, Springer-Verlag, 1982.
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R. Bott and L.W. Tu: Differential Forms in Algebraic Topology. SpringerVerlag, New York, Heidelberg, Berlin, 1982.
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R. Bott and L. Tu: Differential Forms in Algebraic Topology, Springer, 1980.
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R. Bott and L.W. Tu: Differential Forms in Algebraic Topology. SpringerVerlag, New York, Heidelberg, Berlin, 1982.
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R. Bott and L. W. Tu: Differential forms in algebraic topology. Springer--Verlag, Berlin--Heidelberg--New York, 1982
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R. Bott and L. W. Tu 1982: Differential Forms in Algebraic Topology, Springer, Berlin and New York.
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