homotopy groups

hi folks! here is my latest opus, “homotopy groups with coefficients.” if you think that anyone else might be interested, it is available for downloading on my website, at the moment listed under the title "samelson products." that will probably change in the near future to "homotopy groups with coefficients."

by sending you this, i reveal that my email list has multiple purposes.

one, it serves to record and disseminate my travels, especially as these relate to cuisine. beef in argentina and fish in chile come to mind.

two, it serves to provide an outlet for political commentary. i am proud of the fact that i have defended canadian health care from political attacks and also that i recognized very early the fact that a major candidate was classic white trash.

but a third purpose is to communicate with some of my old friends in the mathematical world. this can cause trouble. one of our former graduate students came up to me at a conference the other day and said that there was a rumour that i had a blog which was available and that in it i had made some criticisms of grothendieck. he seemed shocked. oh well, it is a good attitude for him to have until he gets tenure.

so now i am utilizing the third purpose. in the words of john harper, “you just finished writing a 565 page book. why in the world are you going over the same ground again?” well there is a reason. let me tell you what it is.

in the process of finishing up the book, one finds many small errors and one corrects them. but, after the process should have been completed but before the publication, i found an error which disturbed me. it was not a typo, it was a real error, albeit minor.

it concerned the effect of the hopf map on the justification of internal properties of the mod p homotopy bockstein spectral sequence. if p was an odd prime, it had an effect on arguments in dimension 3. if p was equal to 2, it effected the arguments in all dimensions. in fact, the description remained essentially the same but the arguments needed to be much more suble. i was resigned to not being in time to change it.

but it gnawed on me. people told me about ancient chinese painters including mistakes on purpose so as to not anger the gods by misguided pretensions to perfection. i felt that my book was already well insulated from charges like that!

but i did communicate my concerns to cambridge, along with what i thought was a brilliant idea. i remembered that the classic princeton monograph by steenrod and epstein used to come with an errata sheet in the back. so i asked whether we couldn’t include one with my book.

up went this request to the highest levels of the editors of cambridge university press. they did something wonderful and totally unexpected. they said, ok, if you really want us to and if you don’t mind the chance that this might delay publication, we will let you make your changes in the body of the text. there will be no errata sheet! i speculate that they felt that the princeton practice of an errata sheet, especially immediately upon publication, was a plebian colonial practice and far beneath them.

so with a few additions to the text, primarily in the form of the words “let p be an odd prime” and “let the dimensions be greater than 3” at the beginning of three sections in the chapter on bockstein spectral sequences, all was fixed.

i had committed two cardinal sins. one, i thought that, if the definitions of the bockstein spectral sequence were the same at all primes, then the general inner workings would be exactly the same. not so! 2 is always different, sometimes in subtle ways. two, even at odd primes, i was a little careless in low dimensions. it turns out that the world of odd primes does not totally part ways with the prime 2 until after dimension three. to the cognescenti, let me just say the words, “the hopf map vanishes at odd primes then, but not before.”

so i did the quick fix of excluding the prime 2 and dimensions less than 4. this effected nothing else in the book. but i knew in my heart that it was a copout, that the prime 2 and dimension 3 both deserved a total rehab. there just wasn’t enough time and space to do that now. maybe there would be time in the second edition, if there is one. we can hope.

but here is the total rehab. it is an expository paper with more depth than it seems. in fact, it contains some new things in the form of significant attention to distributivity laws in low dimensions. this is used to show that certain maps, while sometimes different from multiplications, have the same images in homotopy. in more detail, it has

1) a better proof of the uniqueness of smash decompositions of moore spaces

2) an understanding of the need to use certain fake multiples of the identity in low dimensions in preference to the true multiples of the identity. at odd primes and in dimensions greater than 3, the fake multiples are the same as the true multiples. these maps look like multiples of the indentity from the point of view of intergral cohomology but they are not homotopic to the true multiples. they have advantages, they desuspend one more time and they are they are really zero when they say they are.

3) after introducing these fake multiples, it requires some rather suble computations with hilton-hopf invariants to show that these fake multiples allow the same old identification of the higher terms in the mod p homotopy bockstein spectral sequence. this problem only occurs in dimension 3 for odd primes but it is in all dimensions at the prime 2.

4) in short, that is the answer to harper’s question of “why?”

let me add some reflections. in the course of dealing with real problems in low dimensions, i was forced to rethink in depth the basic foundations of my subject. one finds better ways of doing things. this does not mean that the previous ways are incorrect. for the most part, they are solid. but one now has a far deeper understanding of how things go, what is clear and what is not. in my experience, it is difficult to achieve this clarity without going through a significant review, a review which is far more intensive than a mere rereading would be. perhaps the opportunity to teach another course on these topics would have been fruitful. in the real world of calculus courses and retirement, this opportunity was not available. the discovery of some mistakes and the blessing of free time in retirement led me to redo some of the foundational parts of the book.

and now let me close by asking you all a question. this paper will live for a while on my website, available for downloading by those who wish to do so. perhaps this is a modern form of publication. but, especially when one has had an aortic dissection, one wonders how permanent it is. does anyone out there know of a possible home for publication of a primarily expository paper like this? being emeritus, i do not need to publish. but i still like to and it seems that there are very few homes for papers like this.

in the old days, there were some avenues which no longer exist. for example, some mathematics departments used to make available to the cognescenti the knowledge that they had a collection of mathematical notes, aka typed reprints. often for a nominal charge, you could buy these things.

notre dame, the university of chicago, princeton, and harvard come to mind as places which carried on this practice. notes of artin on galois theory, notes of halmos and kaplansky on naive set theory, notes of bott on k-theory, notes of milnor and stasheff on characteristic classes, and notes of milnor and moore on hopf algebras come to mind as works which originated in this way. all of these are classics. after many years, some of these notes eventually found their way to become books published by mainstream publishers.

but the original notes sometimes had a freshness and spontaneity which their descendents in their full maturity might lack a little. i still think that the early version of characteristic classes is a delight to read. it gets you quickly into the heart of the stuff with beauty and no fuss. who cares if it does not give a rigorous description of the attaching maps for the shubert cells in a grassmann variety? there is nothing wrong with reducing matrices to row echelon form! everybody knows how to do that. or should know. the mere fact that this just gives the open cells should not bother people at first reading. the important ideas are there.

and the original version of the milnor-moore paper on hopf algebras, before they polished it up, is a lot easier to read than the final annals paper.

but those things are all gone now.

let me tell you the tale of another expository paper i wrote. that one was on new theories of homotopical localization and applications to old questions of serre on the infinity of nonvanishing homotopy groups of a simply connected nontrivial finite complex. i am pleased to say that one of the founders of this new localization, a certain israeli topologist many of us know well, used the notes in his course and told me that his students liked the notes and found them illuminating.

i thought i had found the pertect home for this paper, a certain swiss journal with an expository slant. so i sent it there. imagine my surprise when the editor informed me that they were rejecting the paper on the grounds that its subject matter was too “arcane to be of much interest.” after expressing my surprise that some of the early work of serre in homotopy theory was now regarded as arcane, i withdrew the paper and found a lovely home for it in the proceedings of a conference which was held in the swiss alps. those proceedings are now available in the contemporary mathematics series of the ams.

i bit my tongue. i did not point out to the editor of that swiss expository journal that they had published a paper on rational homotopy theory devoted to a sub area that some might think of as arcane. i did not say to the editor that i would communicate to serre the editor’s opinion that serre’s early work was arcane. and i did not communicate it. except for some mild sarcasm, i was exceptionally polite. i withdrew the paper without argument.

but i formed a low opinion of that editor and i will never again send a paper to that swiss journal! the quandary is that one is reluctant to impose too much on the kindness of editors of conference proceedings. so where does one send something like this paper? any ideas?

best wishes,

joe n