We thank all colleagues who pointed out mistakes and misprints. Any more hints are welcome, do not hesitate to send me an email!
Known mistakes so far:
p. 6 ff: The volume form should be defined with a different normalisation, thus leading to several corrections:
l.-9: Put a factor $(-1)^q$ in front of the determinant.
l. -4: The volume form should be defined as $\sigma_1\wedge\ldots\wedge\sigma_n$ independent of the signature, i.e. the factor $(-1)^q$ should be discarded.
p. 7, l. -9 and -11: Again, remove both factors $(-1)^q$.
p. 8, l. 2,3: remove the signs $(-1)^q$ and $(-1)^{2q}$ up front, respectively. Notice that the final result, l. 4 remains the same!
p. 15, l. 8: the $dx^i$ on the right hand side should be $dx^j$.
p. 30, l. 7 and 8: it should be twice $dx1\wedge\ldots\wedge dx^k$ (i.e., the upper index $n$ should be $k$).
p. 24, l. -10: This formula requires a proof, of course (not difficult). Since it's a useful one, I guess we should have stated it as a lemma.
p. 31, proof of Thm 7: same thing, every $dx^n$ should be replaced by $dx^k$.
p. 32, l. 11: Integrate over $[0,1]^{k+1}$ instead of $[0,1]^k$.
p. 33, l. 3 and 4: before the last integral sign in l.3 and the first integral sign in l.4, a factor $(-1)^{j+\alpha}$ is missing.
P. 35, l.3-4: The argument is right, albeit a bit obscure. The implicit function theorem is not needed, one just observes that $A$ cannot carry $(k+1)$-forms, and that's it.
p. 51, condition (2) of Thm 2: replace "bijective" by "injective".
p. 58, l. -1: the function "g" in this formula should be "h" (there is no function called g here).
p. 127, l. 13-15: in a), there is a typo, the equation should read $(e^{2t} - x^2) + x dx/dt =0$. b) ist ok, c) should be deleted (not integrable by any elementary means).
p. 127, l. -3: Add the initial condition that $A$ should coincide with a given matrix $A_0$ for $t=0$, $A(0)=A_0$.
p. 150, Theorem 13 (2): the second $V_1$ in the subscript should be $V_2$.
p. 184, Theorem 42 (2): the second $V_1$ in the subscript should be $V_2$.
p. 178, l. 7/9: replace "meridian" by "parallel circle" (translation mistake).
p.223, l. 16: "and equality holds if $G$ is connected" (translation mistake).
p. 226, exercise 5: Add assumption that G is connected.
p.227, exercise 8 b): replace fraction by $\frac{az+b}{\bar{b}z+\bar{a}}$ (the coefficients in the denominator should be interchanged).