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**Extra resources for Cobordisms and spectral sequences**

**Sample text**

Let F be a space, let B p be a manifold (connected for simplicity), and let Bz ! B denote the universal cover. B/ ! B/ determined by deck transformations on Bz and by ' on F . The action on Bz F is free and properly discontinuous, hence M is a foliated space. It is foliated by leaves `x which are the images of Bz fxg as x 2 F . There is a natural map M ! B and the composite `x ! M ! B is a covering space. F / then M is a smooth manifold. A very special case of the above construction is of considerable importance.

This shows that g 2 C 1 and g 2 N , and the theorem is proved. It remains to construct the gk . Put g0 D f ; then the hypotheses are true vacuously. Suppose that 0 < m and we have maps gk 2 N , 0 Ä k < m, satisfying the inductive hypothesis. Um ; Vm / j h D gm 1 on Um Wm g: Defi ne T W Ᏻ ! h/ D h gm 1 on Um , on X Um . N / is nonempty. S Form the closed subset K D kÄm Ck \ Um of Um . Then gm 1 W Um ! Vm is C 1 on a neighborhood of K Wm . Um ; Vm /. We conclude that the maps in Ᏻ which are C 1 in a neighborhood of K are dense in Ᏻ.

X; / by multiplication of measures by functions with the left and right actions being the same. I. 5. The class of locally traceable operators is closed under adjoints and is a two-sided module over Ꮽ. 1 Proof. We have already seen that the locally traceable operators are closed under adjoints. To see that this class is a two-sided module over Ꮽ, it suffi ces, by taking linear combinations, to show that gP is locally traceable when P is nonnegative locally traceable, and g 2 Ꮽ. gP Pg / are locally traceable.