# Some possible mistakes in Bott and Tu

I've been reading Bott and Tu's book "Differential Forms in Algebraic Topology" and it seems that this book has an enormous list of errors. Therefore I would like to confirm if the parts that I will cite are, in fact, wrong and, if I'm correct, I would like to know how to solve these problem. Here is a link to the edition of the book that I'm using http://www.maths.ed.ac.uk/~aar/papers/botttu.pdf.

1) In page 43 (proof of theorem 5.1), they claim that finite intersections of geodesically convex open sets are geodesically convex. However choosing two open sets covering the circle, would give a counterexample to this. Maybe the correct notion of good cover (in page 42), should be a covering such that finite intersections are diffeomorphic to finite disjoint copies of $\mathbb{R}^n$. (As noted by Kevin Carlson and Mariano Suárez-Alvarez the geodesics must be minimizing, therefore I was wrong)

2) In page 65, they define a sequence of vector bundles to be exact if they're fiberwise exact. This is not the correct definition, since the kernel of a morphism that collapse some fiber will be $0$.

3) In page 69, there is a diagramm and they claim it's commutative, but there's no reason at all. And just above the diagram they say: "If $f: M \rightarrow M'$ is an orientation preserving map of oriented manifolds, $T$ is a tubular neighborhood of the closed oriented sub manifold $S$ in $M$, and $f(M')$ is transversal to $S$ and $T$, then $f^{-1}T$ is a tubular neighborhood of $f^{-1}S$ in $M$.". However $T$ has always codimension $0$, so how can $f(M')$ not be transversal to $T$, clearly, there's something wrong going on here. Furthermore there is no reason (beyond the intuition) to $f^{-1}T$ be a tubular neighborhood of $f^{-1}S$. So this part, clearly, needs some clarifications.

4) In page 70, execise 6.32 may not be right for any deformation retract, since the composition $(ir)^* \in \text{End}(\Omega(\mathbb{R}^n\setminus \{0 \}))$ (for some retraction $r$ and an inclusion $i$) is not the identity. Therefore $dr\wedge r^{*}(d\theta|_{S^2})$ may not be positive, for instance.

5) In page 79 (just before the beginning of the section 7), they claim that $s\pi: E_0 \rightarrow E$ is clearly homotopic to the inclusion $i: E_0 \rightarrow E$, where $E_0$ is the complement of the zero section, $s$ is the zero section and $\pi: E \rightarrow M$ is the vector bundle. This looks like very weird. For instance, consider $M = *$ (a point), then they are claiming that the map $\mathbb{R}^n \setminus \{0\} \rightarrow \{0 \}$ is homotopic to the inclusion $i: \mathbb{R}^n \setminus \{0 \} \rightarrow \mathbb{R}^n$, so the sphere would be homotopic to a point! (As noted by Kevin Carlson, in fact, this is equivalent to the contractibility of $\mathbb{R}^n$, therefore I was wrong. I don't know why, but I was thinking about isotopies fixing the circle)

6) In page 102 and 103, they claim that the collating formula (page 102) is a morphism $f: C(\mathcal{U}, \Omega) \rightarrow \Omega(M)$ that induces an isomorphism in level of cohomology by using a a homotopy operator (in the beginning go the page 103). However the collating formula is not well defined since in the second sum $\sum_{i=0}^{n+1}K(-D''K)^{i-1} \beta_i$ the term $K(-D''K)^{-1}$ is not well defined. I tried to exclude this term, but then everything goes wrong (for instance, the first computations in page 104 does not hold).

7) In page 104 (proposition 9.8), they claim to have created an explicit isomorphism between \v{C}ech cohomology and de Rham cohomology $f(\eta) = (-D''K)^n \eta$ by using the collating formula. However as stated above (in 6), the collating formula is ill defined, so this explicit isomorphism may not be true. However, in page 120-122, they use this isomorphism to prove that the Euler class and the global angular form coincides with their previous definition. So, again, this may not be right, since the collating formula was ill defined.

8)In page 108 (example 9.14), they claim that for $M = \mathbb{Z}^{+}$ and $F = \mathbb{Z}^{+}$ the Kunneth formula does not hold by claiming that $H^{0}(M) \otimes H^{0}(F)$ consist of finite sums of integer matrices of rank one. However $H^{0}$ is contravariant, so $H^{0}(F) = H^{0} (M) = H^{0} (\mathbb{Z}^+) = H^{0} (\coprod^{0 <i < \aleph_0} *) = \prod_{0<i< \aleph_0} \mathbb{R}$, hence $H^{0} (M) \oplus H^{0} (F)$ is not isomorphic to $\bigoplus_{0<i< \aleph_0}\mathbb{R}$ as they claim.

9)In page 109, the definition of a constant presheaf is wrong. They define a presheaf $\mathscr{F}$ to be constant with abelian group $G$ if for every connected open set $U$ and any open set $V \subset U$ the restriction $\mathscr{F}(U) \rightarrow \mathscr{F}(V)$ is the identity. This is clearly wrong, since when $V$ is disconnected $\mathscr{F} (V) = G^{\pi_0 (V)}$ when $\mathscr{F}$ is a constant sheaf. The correct definition should be that $\mathscr{F}(U) \rightarrow \mathscr{F}_x$ is the identity for every connected set $U$ and $x \in U$. (As noted by Kevin Carlson, they intended to define just a pre sheaf. I thought that they wanted to define a constant sheaf without talking about equalizers. Anyway, the requirement for $U$ to be connected still weird, since there would be no locally constant presheaves on a totally disconnected space, but this is just an unfortunate choice made by the authors)

(I've double checked the book and they really use the constant SHEAF instead of the constant presheaf as they've defined. There are a lot of occurrences where this is problematic. For instance, in page 189 (Remark), they claim that the zeroth \v{C}ech cohomology has dimension equal to the number of connected components. Furthermore, in page 97, they define the constant "presheaf" $\mathbb{R}$ as the sheaf of locally constant functions. So, clearly, there's a problem here)

10)In page 133 (remark 12.4.2), they claim that a section of an oriented vector bundle $s^{*}: H_{cv}(E) \rightarrow M$ ($s: M \rightarrow E$) must factorize through the inclusion $i: H_{cv} (E) \rightarrow H(E)$. However this is not always true, since $i$ may not be injective.

11)In page 135 (proof of proposition 12.8), they claim that $\int_{S_x}\Phi = \int_{E_x} \Phi$ because $E_x$ is homotopic to $S_x$ modulo the region in $E$ where $\Phi$ is zero. However this makes no sense, since the integral is not invariant under homotopy of the domain. Although homotopic maps have the same integral.

12)In page 140 (exercise 12.12.1), the index of the sum should start at $0$ and not at $1$.

13)In page 147 (the extension principle), they claim that a map $\partial I^k \rightarrow X$ may be extended to $I^k \rightarrow X$ if $\pi_q (X) \cong 0$ for $q \leq k-1$. However they provide no references for this result and, since this book has so many errors, I'm starting to doubt if this is true.

14)In page 177 ("The Gysin Sequence"), they claim that in an oriented sphere bundle every locally constant pre sheaf has no monodromy and, therefore, is constant. However I could not see why this is true (by using the already introduced machinery). I think they're confusing it with the fact that a locally constant presheaf in an oriented vector bundle is constant (by using the argument in page 131-132 that restrict the Thom class). Maybe it's possible conclude this for sphere bundles coming from vector bundles (by using the fact for vector bundles), however not every sphere bundle may have its structure group (as a $G$-bundle) reduced to $O(k)$.

15)In page 178 (in the bottom of the page), they claim that an element of $H^{n-k}(M) \otimes H^k(S^k)$ may be represented by $\pi^{*}\omega \wedge (-\psi)$ where $\omega \in H^{n-k} (M)$ and $\psi$ the global angular form. However this is not, in general, true since Leray-Hirch theorem does not hold in this case and $\psi$ is not closed (though it's locally closed).

16)In page 183, the face operator is not defined correctly. The correct definition should be $\partial_q^i (\sum_{j = 0}^{q-1} t_j P_j) = \sum_{j = 0}^{i-1}t_jP_j + \sum_{j = i+1}^{q}t_{j-1}P_j$.

17)In page 189 (in the bottom of the page), the exact sequence for singular cohomology is wrong. Instead of direct sums, it should be products since the functor $\text{Hom}(-, \mathbb{Z})$ is left exact (in the opposite category), hence it sends colimits to limits.

Sorry for the long list. If someone know about any other possible mistakes in the book, please, let me know.

EDIT I've checked the new edition of the book and some mistakes were corrected. More specifically, 6, 7, 9, 12, 16 and 17 were corrected. I should mention too that, as commented above, 1, 5 and 13 are not mistakes.

• You do not really mean that the book «has an enormous errata» but that it has lots of errors. An errata is a list of corrections, not the errors themselves (the word erratum, whose plural is errata, does mean error too, but it is rarely used in that sense in this context, and then it is not enormous :-) ) – Mariano Suárez-Álvarez Jun 25 '15 at 6:54
• People do say that, that is true. They are misusing the expression :-) – Mariano Suárez-Álvarez Jun 25 '15 at 7:03
• I think in (11) they mean 'homologous', so as to apply Stokes' Theorem. – Sir Wilfred Lucas-Dockery Sep 3 '16 at 22:36
• @user40276 If you look at the diagram in the book (where conveniently Z is $0$ dimensional and the fibres are lines), $E_z$ and $N_z$ are conspicuously the sides of a 2-simplex, with the third side being outside the support of the Thom class. Using the orientation assumptions, can't you say $0=\int_{\Delta}{d\Phi}=\int_{E_z}{\Phi}-\int_{N_z}{\Phi}+0$? This is just an attempt and I'm hoping to make it rigorous in general. I hope this is of some help to you! – Sir Wilfred Lucas-Dockery Sep 4 '16 at 18:43
• Don't they? If $E$ has fibres of dimension $n$ and $M$ is a manifold of dimension $d$, then $E$ as a manifold has dimension $n+d$. Since we are given a transversal section in $E$, we know that $d+d \geq n+d$. Also, the dimension of $Z$ would then be $d-n$. So the co-dimension of Z in M is $d-(d-n)$, which is $n$! – Sir Wilfred Lucas-Dockery Sep 8 '16 at 17:42

I've checked five of your claimed errors, none of which you're right about. There are other cases in which you've found obvious typos, which are admittedly annoying but which shouldn't cause any real trouble.

It may be you've found some significant errors in the other points which are more complicated for me to check, but based on the following evidence it looks like you need to take the authors more seriously and check your objections more critically. As a bit of unasked for advice, I should point out that Bott and Tu is one of the most important and valuable books in the subject, and it seems likely that you're losing something in making your reading of it a contest with the authors. On the other hand, it does seem to be leading to you reading more closely than many would, so perhaps there's no harm done.

1) You need to think more carefully about what geodesic convexity is for the circle.

2) The kernel of a map of vector bundles collapsing one fiber is not a vector bundle. Given that your sequence is a sequence of vector bundles, this is fine.

5) No, they'd be claiming the inclusion $\mathbb{R}^n\setminus\{0\}\to \mathbb{R}^n$ is homotopic to a constant, which is obviously true since every map to $\mathbb{R}^n$ is.

9) No, there's a difference between a constant sheaf and a constant presheaf.

13) This is something you can easily check for yourself. For instance when $k=2$ it amounts to claiming that I can extend every map from a circle to a map from the disk when my space is simply connected. Indeed, a nullhomotopy of a map from the circle is exactly the same thing as a map from the disk.

• Thanks for your answer. Can you justify 1? Choosing two connected open sets covering the circle, will give two disjoint neighborhoods. About 2, the kernel of a map collapsing just one fiber is $0$ (the zero vector bundle). – user40276 Jun 25 '15 at 7:00
• Are those two open sets geodesically convex? – Mariano Suárez-Álvarez Jun 25 '15 at 7:04
• @MarianoSuárez-Alvarez I'm not sure of myself, but it looks like that connected open sets are always geodesically convex in the circle. For instance, consider $U$ the open set that varies the angle from $0$ to $\pi$. – user40276 Jun 25 '15 at 7:07
• Look in the wikipedia page for «geodesic convexity». – Mariano Suárez-Álvarez Jun 25 '15 at 7:09
• The problem is that picking kernels in the category of quasi-coherent sheaves (and, now I'm thinking about algebraic geometry, so it's just for the complex case) is not the same as picking kernels in the category of vector bundles (which, by the way, is not abelian). Picking stalks is exact, however picking fibers (in the sense of algebraic geometry) is NOT exact ( it does not preserve finite limits) because tensoring by the residue field is NOT flat. – user40276 Jun 27 '15 at 0:18