The following paragraph is taken from the original “too personal for publication draft” of an article entitled ” ‘Final values’ of functors” by Shmuel Weinberger for a volume in honor of Guido Mislin’s retirement from ETH. (L’enseignement Mathematique 54(2008), 180-182.) Shmuel’s remarks about making conjectures and the different types of conjectures appear here for the first time.

**“Final Values” of Functors**

** **

Shmuel Weinberger

A personal letter for, and to, Guido Mislin, a mathematician who always epitomized for me elegance, taste, and precision.

Guido Mislin

I have, on very rare occasions, conjectured things I really believe with evidence by analogy and the religious belief in the essential simplicity of the mathematical universe (looked at correctly)[1]. Besides beauty, such a conjecture[2] pragmatically serves as a guide through unknown landscape. And even an unreliable guide helps to point in the right direction. In fact, an unreliable guide known to be unreliable can be useful indeed. Sometimes one makes conjectures knowing them to be false, but feeling that their falsity is a deep phenomenon and most of the predictions made with the conjecture as guide will be true[3]. On other times, I have conjectured to lay down the gauntlet: “See, you can’t even disprove this ridiculous idea.” [4] On yet other times, the conjectures come from daydreams: it would be so nice if this were true[5]. And, yet on others one makes a conjecture hoping to probe the landscape that other conjectures have already illuminated[6].

The conjecture (and speculations) that I would like to present to you, Guido, is motivated by ideas I learnt from Goodwillie and Weiss, but I think it is also much in the spirit of the way you sometimes approach mathematics and it is somewhere between the last two kinds…

[1] These include the package of conjectures about homology manifolds made in [BFMW], which incidentally flatly contradict other standard conjectures like the Bing-Borsuk conjecture.

[2] Here I have in mind things like the Riemann hypothesis, indeed huge swaths of number theory; in topology and geometry, Thurston’s geometricization conjecture, Baum-Connes conjecture and Novikov conjectures, and so on are examples.

[3] Here I am thinking of the equivariant version of the Borel conjecture, or the stratified version. See [W, Chapter 13].

[4] I think this is what Lott had in mind (although he was usually careful to call it a question) — but I have less concern for my reputation, and am willing to conjecture the opposite of what I believe -when he formulated the zero-in-the-spectrum problem for all complete manifolds. And it was disproved in [FW]. On the other hand, my dear friend and mentor Donald Newman made a conjecture of this same sort once in rational approximation, only to have it proven by V.A.Popov.

[5] This must be the case for, say, Gromov’s questions about large Riemannian manifolds [G] or the Weinberger conjecture discussed first in print in [D]

[6] The zero in the spectrum problem for universal covers of aspherical manifolds or even uniformly contractible manifolds, also discussed by Lott, is of this sort. The conjecture of rationality of the difference of twisted APS invariants for homotopy equivalent manifolds was made after realizing that it would follow from the Borel conjecture for torsion free groups. By now, it’s been proven three times. [FLW][K][GHW] and [HR].

As further afterthoughts Shmuel writes:I realize that I haven’t given enough thought to the affects of the different kinds of conjectures as they get lives of their own, over time. Also, I have treated all conjectures as if they were good things, when, in truth, I think some conjectures drain the surrounding areas of attention that they more importantly deserve.

References mentioned in the footnotes:

[BFMW] Bryant, Ferry, Mio, and Weinberger, Topology of homology manifolds. Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 324–328.

[D] Dranishnikov, Dimension theory and large Riemannian manifolds. Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 423–432

[FLW] Farber, Levine, with an appendix by Weinberger, Jumps of the eta-invariant. Math. Z. 223 (1996), no. 2, 197-246.

[FW] Farber and Weinberger, On the zero-in-the-spectrum conjecture. Ann. of Math. (2) 154 (2001), no. 1, 139-154.

[GHW] Guentner, Higson, and Weinberger, The Novikov conjecture for linear groups, Publ Math d’IHES, 101 (2005) 243-268

[G] Gromov, Large Riemannian manifolds. Curvature and topology of Riemannian manifolds (Katata, 1985), 108–121, Lecture Notes in Math., 1201,* *Springer, Berlin, 1986.

[HR] Higson and Roe, K-homology, assembly, and rigidity theorems for relative eta invariants (preprint 2008)

[K] Keswani, Relative eta-invariants and C*-algebra K-theory. Topology 39 (2000), no. 5, 957-983.

[L] Lott, The zero-in-the-spectrum question. Enseign. Math. (2) 42 (1996), no. 3-4, 341-376.

[M] Mislin, Equivariant K-homology of the classifying space for proper actions. Proper group actions and the Baum-Connes conjecture, 1-78, Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, 2003.

[W] Weinberger, The topological classification of stratified spaces, University of Chicago Press 1994

William Browder, Sylvain Cappell, Shmuel Weinberger, and Stanley Chang

See this interesting related blog post by Igor Pak. What if they are all wrong? | Igor Pak’s blog (wordpress.com)

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Here is Mike Saks’ take on what a conjecture is: https://youtu.be/hYfeRtdj7cw?t=7m32s

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See this very interesting post by Igor Pak https://igorpak.wordpress.com/2020/12/10/what-if-they-are-all-wrong/

Pingback: Igor Pak: What if they are all wrong? | Combinatorics and more