by Fabian Hebestreit and Markus Land
Early interactions
We both became disciples of Andrew Ranicki during our early education on surgery theory: in the case of the first author directly through lectures on the topic by Andrew, and in the case of the second author mostly by proxy Wolfgang Lück, motivated by the relation between surgery theoretic and index theoretic approaches to the Novikov conjecture. As usual, our first interactions with Andrew were little chats at conferences and summer schools; however, our communication intensified over time. We both had many email exchanges and video calls with him, seeking input on our work, trying to decipher some writing of his or looking for a reference from the golden days of geometric topology. These interactions gave us many fond memories: for a good couple of months when the second author was a postdoc in Regensburg, the first thing to do in the morning was to check Skype and have a brief video call with Andrew about mathematics and also life in general. At almost all times of day, his replies to any and all approaches would be almost immediate, irrespective of whether he was supposed to be otherwise engaged. For example, he would occasionally answer, complete with an accompanying image or video snippet he thought amusing, while in the audience of seminars or conference talks. In addition to answering on the spot, Andrew’s replies were always extremely friendly and inviting, sometimes to his own detriment. Here is an amusing instance of this. Once, in the middle of the night, we asked him (amongst other things) via email about a supposedly missing finiteness condition in the definition of geometric normal spaces in one of his books. Having missed the answer to our question in his almost instantaneous reply, we unrelentingly repeated the question several times, admittedly after a light ale or two, in ever larger fonts. Instead of calling out our misdemeanour, Andrew politely engaged in this “exchange” for the better part of an hour before finally giving up on us for the first (and to the best of our memory only) time. Shortly after, apparently not fed up with our antics, he invited the two of us to visit him and his wife Ida at their home in Edinburgh in February 2018, a meeting about which we shall say more later.
Andrew’s use of Poincaré chain complexes in surgery theory provided what immediately seemed to us the most natural way to capture signature-like invariants in topology, such as the surgery obstruction of a degree one normal map and Andrew’s symmetric signature of a closed oriented manifold, were it not for the long and intimidating explicit formulae that Andrew was so fond of. We urge the incredulous reader to pick five pages from his book Exact Sequences in the Algebraic Theory of Surgery at random to see why it earned the affectionate nickname “the phone book”, yellow cover aside. While studying his work, many times the last resort was to write him an email with a call for aid, which in his characteristic manner he would often enough grant almost immediately, as alluded to above. Full of energy he would explain the meaning underlying his formulae only to then resist any attempt at recasting them in more conceptual terms.
With time it became possible for us to give our own proofs of several of his results, in a framework of L-theory first suggested and studied by Jacob Lurie, which allows for much smoother treatments in many places (at least to our minds). Upon extracting explicit formulae from said framework (and neglecting signs) one would of course find exactly what Andrew had written down decades earlier. His ability to guess and interpret explicit chain-level constructions a priori, however, remains a mystery to this day. His guiding hand and his eagerness to pass on his intuition in this subject is sorely missed. To this day, we find results in his writing that seem closely related to questions we have, or to results we have obtained, if only we could easily peer through his formulae.
How his work influenced ours
Probably one of Andrew’s most prominent results is the identification of the geometric surgery exact sequence for topological manifolds with his algebraic surgery exact sequence based on the assembly map in quadratic L-theory, which in particular paved the way for attacking Borel’s rigidity conjecture for aspherical manifolds (predicting that any homotopy equivalence between such is homotopic to a homeomorphism). As part of this program, Andrew showed that L-theoretic Poincaré duality sets topological manifolds apart from general Poincaré complexes. He ultimately used this rather subtle observation to develop the total surgery obstruction — a single invariant associated to a Poincaré complex which vanishes if and only if it is homotopy equivalent to a closed topological manifold (as long as the dimension is at least 5, and along with a relative version detecting homeomorphisms among homotopy equivalences). His foundational insights in this direction are probably most succinctly summarised by what we have come to call Ranicki’s signature square, that is, \[ \begin{CD} \Omega^\mathrm{STop} @>{{\sigma^\mathrm{s} }}>{{}}> \mathrm{L}^\mathrm{s}(\mathbb{Z})\\ @VV\mathrm{fgt}V @VV\mathrm{fgt}V\\ \Omega^{\mathrm{S}\mathrm{G}} @>{{\sigma^\mathrm{n} }}>{{}}> \mathrm{L}^\mathrm{n}(\mathbb{Z}) \end{CD} \]
where \( \Omega^\mathrm{STop} \) denotes the oriented topological cobordism spectrum, \( \Omega^{\mathrm{S}\mathcal{G}} \) denotes Quinn’s oriented normal cobordism spectrum, and the right-hand terms are the symmetric and normal L-spectra of \( \mathbb{Z} \) analogously defined by Ranicki using cobordisms of chain complexes with duality in place of manifolds. Let us explicitly mention that the right-hand objects evolved with Andrew’s work and consequently there are a multitude of different versions of these spectra. This is a fantastic source of confusion and will become relevant below; in fact, we managed to get ourselves rather confused yet again while writing this very note. The spectra above are (hopefully) denoted \( \mathrm{L}^{\cdot}(\mathbb{Z}) \) and \( \widehat{\mathrm L}^\cdot(\mathbb{Z}) \) in Andrew’s Algebraic L-Theory and Topological Manifolds. Regardless, the top horizontal map in the above square is the famous Sullivan–Ranicki orientation, which takes a manifold to its symmetric signature, essentially its (co)chain complex together with its Poincaré duality equivalence (Sullivan had done something similar away from the prime 2 earlier using real topological K-theory instead of symmetric L-theory). It implements the L-theoretic Poincaré duality for oriented topological manifolds mentioned above, and therefore provides the fundamental class in \( \mathrm{L}^\mathrm{s}(\mathbb{Z}) \)-homology of oriented topological manifolds. In particular it lifts the symmetric signature of a topological manifold along the assembly map in symmetric L-theory and the total surgery obstruction measures exactly whether such a lift exists for a general Poincaré complex. The lower map assigns to one of Quinn’s normal spaces its normal signature and provides a fundamental class in \( \mathrm{L}^\mathrm{n}(\mathbb{Z}) \)-homology for all oriented Poincaré complexes via the map \( \Omega^{\mathrm{SP}} \rightarrow \Omega^{\mathrm{SG}} \) from oriented Poincaré cobordism.
The total surgery obstruction now measures exactly whether this normal fundamental class can be lifted to a fundamental class in symmetric L-homology, and in the relative case of a degree 1 normal map among manifolds, the invariance of the normal signature under normal (and not just Poincaré) cobordisms allows one to extract a class in \( \mathrm{L}^\mathrm{q}(\mathbb{Z}) \)-homology (the quadratic L-spectrum \( \mathrm{L}^\mathrm{q}(\mathbb{Z}) \) is the fibre of the right-hand vertical map) whose image under the assembly map is the surgery obstruction. This is the inroad that connects the algebraic surgery sequence to the topological one.
Many years later and in somewhat vague form, Hesselholt, Madsen, Williams and others conjectured a relation between L-theory and algebraic K-theory — linked by what is referred to variously as real algebraic K-theory, hermitian K-theory, or Grothendieck–Witt theory (the latter is the terminology we adopted). It is in our work on these conjectures that Andrew’s teachings really paid off: over the last two decades the geometric cobordism spectra appearing in the signature square above have been refined to great effect by the combined efforts of Galatius, Madsen, Randal-Williams, Tillmann, and Weiss to what is known as cobordism categories, and we (along with our collaborators Calmès, Dotto, Harpaz, Moi, Nardin, Nikolaus and Steimle) realised that one could use Andrew’s ideas to produce algebraic cobordism categories that similarly refine his L-spectra. Using the method of algebraic surgery he pioneered for the analysis of L-groups, we showed that these algebraic cobordism categories precisely give rise to Grothendieck–Witt spectra. Just as algebraic surgery is an imitation of geometric surgery at the level of chain complexes, we could then transport well known geometric techniques connecting cobordism categories and cobordism spectra into the algebraic world, and connect Grothendieck–Witt and L-spectra. This culminated in a general fibre sequence of the form \[\mathrm{K}(R)_{\mathrm{hC}_2} \xrightarrow{\operatorname{hyp}} \mathrm{GW}^{\mathrm{s}}(R) \xrightarrow{\operatorname{bord}} \mathrm{L}^\mathrm{s}(R).\] A word of warning is in order here. As indicated earlier, Andrew gave two competing definitions of symmetric L-groups for general rings, one 4-periodic and one not generally so. After establishing that these do not differ for the integers, at least in nonnegative degrees, he eventually favoured the periodic version for its supposed simplicity. In Lurie’s formalism, on which our work is based, both of Andrew’s definitions have natural and distinct meaning. It turns out, though very much not by definition, that it is the original nonperiodic version which appears in the fibre sequence above; to distinguish them we usually denote this version by \( \mathrm{L}^{\mathrm{gs}}(R) \) outside this note. This nonperiodic version was far less known outside Ranicki’s inner circle, and perhaps this mismatch explains why conjectures about the relation between K- and L-theory had previously remained vague. Furthermore, the nonperiodic L-groups are only really defined in nonnegative degrees, and in his early works Andrew noted that they can be spliced in an ad hoc manner with quadratic and symplectic L-groups to extend certain long exact sequences also into negative degrees. To the best of our knowledge, quite which long exact sequences he meant was never made explicit, but our set-up canonically provides a definition of negative L-groups as well. As a typical instance of Ranicki’s magic ability to guess formulae, these turn out to be exactly the quadratic and symplectic L-groups. Moreover, this identification has very surprising consequences: for example, it implies that the negative symmetric Grothendieck–Witt-groups identify with quadratic and symplectic L-groups, and are not directly related to the (periodic) symmetric L-groups even in the case of the integers.
At any rate, using this new relation between Grothendieck–Witt and L-theory, we were able to almost completely determine the quadratic and symmetric Grothendieck–Witt groups of number rings and resolve periodicity conjectures of Karoubi — in fact, these conjectures translate exactly to statements in L-theory Andrew had provided decades ago, when comparing his two competing definitions of symmetric L-groups. Unsurprisingly, a plethora of open questions and ongoing projects remain, not least of which is a refinement of his signature square to the level of cobordism categories. We would have loved to hear Andrew’s thoughts about all of them.
What we have described above is certainly not the only point where our work intersected Andrew’s, but we hope to have conveyed an impression of the feeling of working under his wing and of the influence he had on our mathematical thinking.
Our last meeting
We told Andrew about early forms of our ideas (in fact as two separate projects, whose overlap and full impact we had not yet discovered), when we visited him and Ida at their home in February 2018. Despite Andrew’s already advancing leukaemia their hospitality was simply stunning, as probably every one of the many mathematicians (and probably nonmathematicians) to stay at their house will have experienced as well.
During our visit we joined Andrew to what we assumed would be a rather routine check-up in the hospital, and the three of us must have made a strange sight, excitedly chatting about mathematics in the cancer ward, with his laughter filling more the just the room we occupied. Unfortunately, the results of the check-up ended up being anything but routine and Andrew died just two days after we left.
That we can no longer share our excitement for the developments in the relation between higher algebra and geometric topology, into our part of which Andrew was kind enough to pour so much of his energy, leaves a hole that has not been filled to this day.
Du fehlst.
