New L 2-invariants of Chain Complexes and Applications

L 2-invariants and morse-smale flows on manifolds

“…It follows from our previous discussion or from [3,4] that the numbers S i (2) (K) are invariants of the homotopy type of the cell complex K.…”

“…Using numerical invariants of free cochain and Hilbert complexes of the manifold M n (see [3,4]), we give an answer to this problem for i = 0, 1, n − 2, n − 1 and 3 ≤ i ≤ n − 4 in the case where dim M n ≥ 6. Thus, the only unsettled case is i = 2 (and n − 3 by duality).…”

Morse functions on cobordisms

“…C n is a free cochain complex over a ring ƒ; then the numbers D r .d 0 /; D l .d n 1 /; D r .d 0 ; d 1 /; D l .d n 2 ; d n 1 /; and D.d i ; d i C1 / are defined for 1 Ä i Ä n 3: The proof that they are invariants of the homotopy type of a cochain complex .C; d / can be found in [18]. The Hilbert complex…”

“…Theorem 6.1 is proved. Estimates for Morse numbers were studied in [1,[4][5][6][7][10][11][12][13][14][15][16][17][18][19], where other approaches were used. In subsequent papers, we shall give the values of Morse numbers for manifolds of other classes.…”

“…In [18], using new homotopy invariants of free cochain complexes and Hilbert complexes of non-simply-connected closed manifolds W n ; n 8; we gave exact values of the i th Morse numbers for 4 Ä i Ä n 4:…”