## Spin Geometry by H. Blaine Lawson Jr. and Marie-Louise Michelsohn – Annotations on the 1989 version of ISBN 0691085420

Hello,

I’m just looking over the book

Spin Geometry by H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Princeton, N.J. : Princeton University Press, 1989.

I’m looking at there 1989 version. It’s  ISBN is 0691085420 /  9780691085425.  There is a newer print from 1994 but that’s in the reference library, only. So maybe all the stuff, I’m writing here is already heavily outdated.

Anyway. While reading the book I come across sections I don’t understand. I want to pose some of my questions here. Furthermore,  I want to list the sentences, which I assume to contain (spelling) errors in some of the formulas. In a way this is my small contribution to a public online errata sheet for the book Spin Geometry.

Page numbers reference the 1989 edition. I’m just reading excerpts from the book, so this list is probably far, far, far from complete. It’s just the four or five things I marked when I read those lines the first time. And: I’m just reading excerpts from the book, so I might be wrong about my “corrections” because I lack the complete picture or previous explanations.

## corrections of spelling erors:

page 16, Corollary 2.6, Proof: It says “Since $N(\phi) = q(v)$ for all $v\in V$” and I guess it’s supposed to be “$N(v)=q(v)$”

page 112:  It should be “Riemannian” instead of “Remannian”.

page 126: There’s a space missing before “Then” within the theorem 5.16, right before equation (5.15).

page 138: I guess equation (6.16) should read $\hat{A}(X\times Y) = \hat{A}(X) \cdot \hat{A}(Y)$ as this is a equation of numbers. (I think I look up that this problem also occurres in edition 2 of the book, so maybe I’m wrong here, dunno)

page 172: Before equation (2.8) it should be noted that $C_1>0$.

somewhere around page 241: I noticed a missing “p”. I can’t find it now. But as far as I remember the text either read “com lex” or “com act”. But I don’t sse it on page 241 anymore. Maybe memories are playing tricks on me.

page  278: “elliptic is an integer” should be “elliptic differential operator is an integer”

page 372:  Example before Remark A1, references “Lemma 1.1”. That’s a reference to “Lemma II 1.1”

## questions:

page 79, equation (1.2), it reads: “From the fibration $O_n \to P_O(E)\to X$, there is an exact sequence. $0\to H^0(X,\ZZ _2) \to H^0(P_O(E),\ZZ _2)\to H^0(O_n,\ZZ _2)\stackrel{w_E}{\to} H^1(X,\ZZ _2)$.” I guess that’s okay. But I wonder, what $O_n \to P_O(E)\to X$ means. I guess there is no canonic morphism $O_n \to P_O(E)$, so how am I supposed to understand this notation exactly. “H^0(P_O(E),\ZZ _2)\to H^0(O_n,\ZZ _2)” is probably well defined as all $O_n$-fibre are homeomorphic and there should be a homotopy connecting them somehow. page 80: There is the same problem here. There are two commutative diagrams following (1.3) and both of them that that there are embeddings $\ZZ _2\to P_{Spin _n}(E)$ and $Spin _n \to P_{Spin}(E)$. I don’t see these embeddings. I see that one can choose one but it’s not canonical. Even if one came up with the idea to choosing a base point the embedding wouldn’t be canonical. As far as I understand it, that’s just the point about principal bundles that there is _no_ canonical embedding. Michelsohn and Lawson do introduce those diagrams by “The diagram of fibrations is…”. So maybe the arrows in the diagram aren’t to be understood as maps. I just don’t know.

page 79, after equation (1.2), it reads: “We can define $w_1(E)=w_E(g_1)$ where $g_1$ is the generator of $H^0(O_n;\ZZ_2)$.” I wonder: What is $H^0(O_n;\ZZ_2)$? Shouldn’t that be: $(\ZZ _2)^{\text{number of connected components of} O_n} = \ZZ _2 \times \ZZ _2$? This is probably wrong, as the book says that $H^0(O_n;\ZZ_2)$ has a special generator $g_1$. Could you tell me where I’m wrong?

page 117, Proof of Theorem (5.7): It reads “…. let d(x) be a regularization of the distance function from [some point] $x_0$.” What’s a regularization? Isn’t the distance function smooth as long as the metric is smooth? At least $d(x)$ has to be continuous due to easy reasoning from Topology I.

page 135, equation  (6.4): It reads: “In fact, the principal symbol of $D^k$ at a cotangent vector $\xi$ is simply Clifford multiplication by $\ii \xi$, that is, $\sigma _\xi (D^k) = \ii \xi \cdot : S^k \to S^{k+1}$ for $k\in \ZZ _2$(6,4) (see Lemma 5.1)”.  I think this is wrong. Even Lemma 5.1 says that $\sigma _\xi (D^2) = – \norm{\xi }^2$. I guess the correct formula should be $\sigma _\xi (D^{2l) = (-\norm{\xi })^l : S^k \to S^k$ and $\sigma _\xi (D^{2l+1) =\ii (-\norm{\xi })^l \xi \cdot : S^k \to S^k$ . In all cases $\cdot$ is a reference to the Clifford multiplication.

page 171: What’s the $U^0$? Is this the interior of $U$, $\text{int }U$?

page 171: The book says: “In fact, the function $\abs{D^{\alpha}u(x)}\left( 1+ \abs{x}\right) ^{\abs{\alpha }}$ is bounded for any $\alpha$.” Why is that?

page 193: Why is it ture that: “Then $u= QPu + Su$, and since $S$ is infinitely smoothing, $\norm{u}_s \leq \norm{QPu}_s + \norm{Su}_s \leq C \left( \norm{Pu}_{s-m} + \norm{u}_{s-m}\right)$. I see that the first step is correct. I wonder about the second step. Do I need for the second step that $S\in “\Psi DO _{-\infty} (F,E)}” \subset \Psi DO _{-m}(F,E)$??

page 199: Why are the three definition of the trace the same? Definition 6.3 features a term $\tr (e^{-tP})$, which is defined. But the definition has two forms. Why is $\int _ trace _x [K_t(x,x)]\dd x = \sum _{k=1}^\infty e^{-\lambda _k t}$. And why is this the same as the usual trace on Hilbert spaces.

page 239: The book says: “We thereby recover the Thom isomorphism in its more conventional form (cf. Milnor-Stasheff [Characteristic Classes, Ann. of math. Studies 76, Princeton University Press, Princeton, 1974 ]). I think the book should tell me which part of the equation the usual Thom-isomorphism is. And I’m surprised that it should use the discbundle.

page 240: The book says: “… Then, as prived in Appendix C, Theorem C.8, there is a natural Thom isomorphism $i_!:K_{cpt}(X) \to K_{cpt}(E)$ of the form $i_!(u) = \Lambda _{-1}\cdot \pi ^*u$ where $\Lambda _{-1} = [\pi ^* \Lambda _\CC ^{even}E, \pi ^* \Lambda _\CC ^{odd}E, \sigma ]$ and where $\sigma$ is defined at each non-zero vector e in E by setting $\sigma _e = e \wedge – (e^*)\llcorner$. Here $e^*$ denotes the dual of e.” It is also denoted $e^\sharp$ via muscial isomorphism. $\sigma _e$ is just the Clifford muliplikation by $e$, when identifying $Cl(V)$  with $\Lambda (V)$. I’d like to see why it’s obvious that the Thom isomorphis uses the Clifford multiplication in its definition.

page 280: Theorem 1.3 contains the equation $\left{\ee ^{\frac{1}{2}c}\hat{\mathbf{A}}(X)\right} [X]$. This is used for $c\equiv w_2(X) (mod 2)$. But doesn’t that mean that different solutions differ by a factor in $e^\ZZ$? But $e^\ZZ \cdot \ZZ \not\subset \ZZ$. What haven’t I understood?

page 281 : What does the symbol $\looparrowright$ mean? Is it a just symbol for an immersion? Is that common? See http://en.wikipedia.org/wiki/Immersion_(mathematics) and for contrast: http://en.wikipedia.org/wiki/Embedding

page 291: The book says “(The only possibilities are $S^1$ when $k=1$ and $S^3$ and $S^3/\ZZ _2$ when $k=3$)”. How does $\ZZ _2$ operate on $S^3$? I guess it’s via multiplication $\id$ and $-\id$.

page  291: Below equation $(3.2)$: Can I visualize the $S^3\subset \mathbb{H}$ action on itself, when looking at $S^3$ as $\RR ^3 \cup \{ \infty \}$. Should I be able to see that via the isomorphism $S^3 \cong Spin _3 \cong SU_2 \cong Sp_1$ Are those differemorphisms or are these even group homomorphisms?

page  292: The text reads: “.. the incariant … can be non.trivial while, of course, $\hat{A} = 0$. How can I see that this is “of course” the case?

page 301 : Text says: “$(S^7\times S^2)\# \Sigma \# \Sigma \cong S^7 \times S^2$” where $\Sigma$ is an exotic 9-sphere and $\#$ denotes the connected sum http://en.wikipedia.org/wiki/Connected_sum . How can this be true? Why isn’t the connected sum with the exotic 9-sphere not changing anything? The book says $\Omega _9^{Spin}\cong \ZZ _2 \oplus \ZZ _2$ would help with that. How?

page 302: The book says: “The product of any manifold with S^2 carries a metric with $\kappa >0$. Consequently, in dimensions >= 6, the presence of positive scalar curvature places no restiction on the isomorphism class of the fundamental group.” Why is that? Is that because there’s some identity like $\pi M^6 \cong \pi (S^2\times M^6)$? At least it probably means that for any $M^6$ there is $N$ with $\pi M^6 \cong \pi S^2 \times N$.

page 307: What’s the LateX command for the Atiyah-Singer-operator (it’s a D which is striked out from the lower left to the upper right.)?  And further more, what does $D_E^2>0$ mean. Does it mean that the spectrum is a subset of $(0, \infty )$?

page 308: The book says: “By passing to a subgroup of finite index we may assume that $\Gamma acts freely and properly discontinuously on$\RR ^m$.” Why is this subgroup having finite index? Would there otherwise be a inifinite group acting on the compact$Y$so that building the quotient would be a problem as the action can’t be properly discontinuous? Is that the point or is there some other reasoning? page 309: Why is the$\hat{A}$-degree defined just for maps$f:X\to Y$with the inverse image of every point being a$4k$-dimensional comapct manifold. I need a reminder. Is this just the necessary condition of a$\hat{A}$-genus to be definted? page 310: Corollary 5,11 Why is this established only here? Why isn’t that part of the theorem above? Isn’t that just (B) and (E) [what’s been left out before]? page 310: Why is it that:$X$enlargeable \Rightarrow$X$is$\hat{A}$-enlargeable? For this to be true one possible way would be to prove that for$\deg f \not= 0 \Ra \hat{A}-degree f \not= 0$. this is equivalent to$\hat{A}-degree f = 0 \Rightarrow \deg f = 0$. So why is$\int  _x f^*\omega =0$whenever$\int  _x f^*\omega \wedge \hat{A}(X) =0$. And what’s the$\wedge \hat{A}(X)$to mean here? Isn’t$\hat{A}(X)$just a number? Or do I have to calculate$f^* \omega \wedge \hat{A}$and than put$X$in there? How would that work? Haven’t got it yet. For this to understand, I’d like to know whether$\hat{A}_0(X)=1$? Is that true? page 310: equation (5.11) Is there a specific meaning in the curved brackets? Why is the$[X]$in line two not applied to the first summand? Is it because$\hat{\mathbf{A}}(X)[X]  = \hat{A}(X)$? The$d$seems not important here, as it is a number anyway. page 310: I need a reminder what$K(\pi , 1)$stand for. Is it a topological space$K$with$\pi _1 (K)=\pi $and$\pi _j (K) = \{ 1 \}  $for$j\not= 1$?? How are these spaces called? page 330: What’s$D$in this case? We have a ONB$(e_1, .. , e_n)$of the \RR -vectorspace T_pX over a point$p$. And I have a unitary basis$(e_1,..,e_n)$over the complex numbers. But it remains that$D = \sum _1^n e_i \bullet \nabla _{e_i}$for$\bullet $the Cliffordmultiplication of$\CC l(X)$. But it looks like$D = \mathcal{D} + \overline{\mathcal{D}}$is the summation of$2n$-summands. How can that be? page 331: How can representation theory be a so potent that it’s able to prove that$\CC l^{p,q}(X)$is non-zero exactly for the pairs$\abs{p+q}\leq n$and$p+q \equiv n ( mod 2)$. That’s impressive. page 335: What’s the Hodge automorphism group? ## “insides”: page 61: How can I visualize the Definition 9.14? It defines the wedge$X \vee Y$and the smash product$X \wedge Y$? This sentence suprised my for a while as I thought it ment$X \vee Y$is called wedge product, but I figure the book just says that it’s called wedge, which wikipedia calls wedge sum. I guess the wikipedia articles explain this somewhat http://en.wikipedia.org/wiki/Wedge_sum http://en.wikipedia.org/wiki/Smash_product http://en.wikipedia.org/wiki/Suspension_(topology) . I was surprised that the wedge sum isn’t written with a$\wedge $. And I still have to understand that reduced suspensions are the left adjoined of the loopspace and I might want to know that in general the smash product isn’t associative (Q\wedge Q \wedge N is undefined without brackets.) But for$S^1 \wedge S^1 \wedge … \wedge manifold$everything should be fine. page 81, Theorem 1.4: For some time, I was confused about this theorem. It states: “The spin structures on E are in natural one-to-one correspondence with 2-sheeted coverings of$P_{SO}(E)$which are non-trivial on the fibres of$\pi $.” Here$\pi: E\to X$is a vector bundle of a Riemannian manifold$X$.$P_{SO}(E)$is the$SO(\rank E)$-principal bundle over$X$, each fibre (over$x$) of the bundle consists of all orthonormal basis’s of$E_x$. I was confused about the requirement that the covering shouldn’t be trivial. But than I realized what that meant, and why it’s obviously the right thing to require.$SO(n)$as two 2-sheeted coverings.$Spin _n$and$SO(n)\times \ZZ _2$. For now I assume that these are the only 2-sheeted coverings. The second covering is called trivial. We want the covering of$P_{SO}(E)$to be that way that in every fibre (which looks like$SO(\rank E)$) is covered in the way that$Spin _n$covers$SO(n)$and not like the trivial covering. And that’s just what is stated in the theorem. page 127: For a moment there I should below equation (5.19), that one should choose eigenbundles of$\alpha ^2$instead of$\alpha $. The book is, however, complete right here as the complete bundle is the one eigenbundle for$\alpha ^2$. But$\alpha $has (at most) two eigenvalues in every point. Those are -1 and 1. So there are potentially (and in reality) two eigenbundles. And that are those we choose. page 172: I was wondering what$D^\alpha $were. But then I understood that all this is really done on$\RR ^n$with a trivial bundle . So the dependence on the choice of the good presentation isn’t relevant here. And it’s not important what the connection is chosen here. And all this is later extended to compact manifolds so one doesn’t have to care about the problems that arise when the manifold as infinite size. Being on$\RR ^n $also explains why the reasoning for (2.8) references (2.4) and (2.6) but lacks a reference to the partition of unity. page 231: When looking at page 231 without reading the site before I was wondering what$p$stands for. And it seems$p(E)$is the total (rational) Pontrijagin class as defined on page 228. page 302: “non-zero degree” I asked myself on p 302 what non-zero degree is. There is a Note on page 303 with a reminder how the degree is calculated. page 308: At first I wondered whether$\diam Y = \infty $might be a problem. But it’s not as$Y$is compact to diameter has to be finite. page 330: This isn’t really helping and it’s not a true example. But I wanted to see what this means in terms of$e$and$\eps $for a simple example like$X=\RR ^2$and$J $is a rotation by$90^°$. So$\RR ^2 \otimes \CC = \CC ^2$. (e_1,e_2) is ONB of$\RR ^2$. On has$Je_1 = e_2$and$Je_2 = -e_1$. Furthermore one defines$\eps = e_1 + i e_2$and$\eps = e_2 – i e_1$And one has$\overline{\eps _1 }$and$\overline{\eps _2 }$. Then$(\eps _1, \eps _2)$is a U(n)-basis for$\CC ^2$. And$(\eps _1, \eps _2, \overline{\eps _1}, \overline{\eps _2})$is a ON-basis for$\RR ^4 = \CC ^2$. …. However, the book seems to say on takes a “orthonormal frame field$(e_1, Je_1)$”. So there is no need for naming an additional$e_2=Je_1$in this convention. ??? How does this fit together? page 331: I was suprised about the fact that$\mathcal{J}(\phi ) = p \phi $for$\phi $a pure “form” in$\CC l ^{p,q}(X)$. But than I understood that the important point is the derivation property mentioned in the text. For every element in the product one get’s a summand and it’s either$\phi $or$0$depending on whether$\mathcal{J}$operates on a factor that’s$\CC $-linear or$\CC $-antilinear when seen as a form. It’s probably the other way round for$\mathcal{H}$## “provisional, private notes for reading it a second time”: page 81, Theorem 1.4: For some time, page 194: Don’t remember anymore, but I guess I wanted the proof of Theorem 5.4 to be 2 lines longer. page 230: I want to understand$\hat{A} $and$Td$better than I do now. Actually I want to understand all of §11 better than I do now. page 241: It’s a pity that all this is done only in the compact case. page 299 : Theorem 4.4 due to Gramov-Lawson A) says for a compact, simply-connected manifold, \dim X > 4: If$X$not spin, then$X$carries a metric with$\kappa >0$. Wow, that’s impressive. I have to understand that further, via is not-been-spin an important fact. Feels like there must be loads of manifolds that are not spin. page 300: “By the basic theory of S. Smale, there exists a smooth function …” I want to read more about that. The book suggests “Lectures on the h-cobordism theorem, Princton University Press., Princeton, 1965” for that. page 307: I want to understand the formula$ch E = \dim E + \frac{1}{(n-1)!}c_n(E)$. page 308: What’s the Splitting Theorem of Cheeger and Gromoll. It’s not mentioned in the index. But it’s a reference given to “The splitting theorem for manifolds of non-negative Ricci curvature, J. Diff. Geom. 6 (1971), 119-128”. page 308: I want to understand equation (5.9) where the index of half of the Atiyah-Singer-Index with values in a bundle is calculated. It contains several transformations containing the$\hat{A}$-form and$ch E$and$[X]$and$\deg (f)$. page 309: Wow,$\hat{A}$-degree of a function is quite a cool definition. Having to prove the existence of spincobordisms in order to see that the degree is weel-defined… . That’s quite impressive. Thanks for looking at this blog post. ## U know other content like this? Are you aware of other blogposts, sites, … that feature similar content that I could link to? Are there websites that collect annotations to books? ## Is there a standard for annotating on-line errata for books? Is there some annotation standard for on-line errata? Maybe there is are RDF-a’s or a MicroFormat to make this page machine readable. The dictionary should understand “book”, “edition”, “year”, “author”, … and items containing “page”, “old context”, “new correction” (maybe “line” or “number of lemma/theorem”), “comment”, “BoolIsCorrection”, “BoolIsJustQuestion”, “BoolIsAddition”. I could than write a blog post specifiying: “There is an error on page xx in book xx, it reads “N(\phi)” it should be “N(v)”. I’d than annotate the sentence. Probably it’s not trivial to manage to find the precise location of$N(\phi)$as it contains LaTeX instead of unicode, but even displaying all comments from all over the web for a single page would be great. I always wonder (when reading a science book) why I can’t see questions (and hopefully answers) other people had while reading just the sentence I’m reading right now. Maybe that’ll come with the spread of e-books. If there were a standard like this a webcrawler could search the web for the all annotations. Maybe rank them by auther if there are too many and display them next to the e-book [or Google Books-like pages]. Surely there might be a problem with “page, line”-annotation in the future, incase e-books get rid of those conventions. But I figure, e-books will look like books for some more years. Anyone willing to discuss this? Comment! [Maybe I should have put this in a seperate blog post.] PS: In the above text, I used some LaTeX commands. Read about them in “lshort”. I also used a locally defined command, namely “\newcommand{\ZZ}{\mathbb{Z}}” and$\ii $for the complex number$\ii \in \CC \$.

PPS: I plan to update this blog post in case I come across further questions or answers or errata.