Research interests

I work on stable homotopy theory and related algebra, particularly the theory of formal groups.

An outline

Homotopy theory is the study of topological spaces and continuous maps, except that we do not distinguish between maps that can be continuously deformed into each other. This study is interesting for a number of reasons.
  1. We can use homotopy theory as an approximation to geometry. For example, we might want to classify closed manifolds up to homeomorphism. This is very hard, but as a first step we can attempt a classification up to homotopy equivalence. In dimension one or two, the homotopy classification is the same as the classification up to homeomorphism or diffeomorphism. In dimension three, one can prove strong statements in the same direction, and even stronger statements are conjectured, although there are some known and well-understood cases in which homotopy equivaence does not imply homeomorphism. In dimension four, there is a very pretty classification of simply connected manifolds up to homotopy equivalence. (However, any finitely presented group occurs as the fundamental group of a closed four-manifold, which implies that in general the homotopy classification problem is undecidable.)

  2. We can use homotopy theory to study geometry in a more indirect manner. For one example of this, we might have two manifolds $M$ and $N$, and we might want to know about the isotopy classes of embeddings of $M$ in $N$. In particular, when $M=S^1$ and $N=S^3$ this is just knot theory. In general, we can define a space $\text{Emb}(M,N)$ of embeddings, and our task is to understand the set $\pi_0\text{Emb}(M,N)$ of path-components in this space. One approach is to write $M$ as a union of simpler manifolds, say $M_1,\dotsc,M_k$ One can then use the methods of homotopy theory to relate $\pi_0\text{Emb}(M,N)$ to the higher homotopy groups of $\text{Emb}(M_i,N)$, and various related spaces.

  3. Another indirect application is Thom's approach to bordism theory. We say that two smooth closed manifolds (say $M$ and $N$) are bordant if there is a manifold W (one dimension higher) whose boundary is diffeomorphic to the disjoint union of $M$ and $N$. We write $MO_k$ for the set of bordism classes of smooth closed manifolds of dimension $k$. One can make these sets into a graded ring, with addition given by disjoint union and multiplication by Cartesian product. As the union of two copies of $M$ is the boundary of $M\times I$, we see that $MO_*$ is actually an algebra over $\mathbb{Z}/2$. Rene Thom showed how to interpret $MO_k$ as a homotopy group of a certain space, and thus how to calculate it using the methods of homotopy theory. His remarkable answer was that $MO_*$ is a polynomial algebra over $\mathbb{Z}/2$ on countably many generators, one in each degree not of the form $2^k-1$. This is striking not only because it is a strong result, but because the form of the answer seems totally unexpected from a geometric point of view. We shall have more to say about this later.

  4. One can also translate some interesting problems in algebra into questions in homotopy theory. A good example is Quillen's definition of algebraic $K$-theory. Let $R$ be a commutative ring. We define $K_0(R)$ to be the set of equivalence classes of expressions $P-Q$, where $P$ and $Q$ are finitely generated projective modules over $R$, and $P-Q$ is identified with $P'-Q'$ if $P\oplus Q' \oplus F\simeq P' \oplus Q \oplus F$ for some finitely generated projective module $F$. One can use the tensor product operation to make $K_0(R)$ into a ring. Next, we define $GL_n(R)$ to be the group of $n\times n$ invertible matrices over $R$. This can be thought of as a subgroup of $GL_{n+1}(R)$ in an obvious way, and we define $GL(R)$ to be the union (or direct limit) of all these groups. We let $E(R)$ be the smallest normal subgroup containing the elementary matrices, and define $K_1(R) = GL(R)/E(R)$. One can check that this group is Abelian. One can also define a group $K_2(R)$ by similar but more complicated methods. There are many clues that these groups should be just the first few of an infinite sequence - for example, there are product maps $K_i(R)\times K_j(R) \to K_{i+j}(R)$ for $i+j\leq 2$, and the groups fit into various exact sequences and so on. However, for many years no one could find a workable definition of $K_i(R)$ for $i>2$. Quillen's approach was to define a natural space $K(R)$ such that $\pi_iK(R) = K_i(R)$ for $i\leq 2$, and then to define $K_i(R)$ to be $\pi_iK(R)$ for $i>2$. This idea was very successful, and the resulting theory has had many applications in number theory and so on.

  5. Another interaction between algebra and homotopy theory involves the theory of classifying spaces of groups. For any group $G$ there is a space $BG$ such that $\pi_1BG = G$ and $\pi_k BG = 0$ for $k\neq 1$, and this space is unique up to homotopy equivalence. An interesting example is the case when $G$ is the symmetric group on $k$ letters. Let $V$ be a vector space of countable dimension over $\mathbb{R}$, and let $B$ be the set of finite subsets $A\subset V$ such that $|A|=k$. If we give $B$ a suitable topology, it turns out that $BG = B$. One can use topological invariants of $BG$ to study the group theory of $G$; for some classes of groups this can be very helpful.
I now give a very brief outline of a method one can use to calculate the bordism ring $MO_*$, as mentioned above. Let $R$ be an algebra over $\mathbb{Z}/2$. We need to consider various sets associated to $R$. \begin{align*} C(R) &= \{ f(x) \in R[[x]] : f(0)=0 \text{ and } f'(0) = 1 \} \\ G(R) &= \{ f(x) \in R[[x]] : f(0)=0 \text{ and } f'(0) = 1 \text{ and } f(x+y) = f(x) + f(y) \} \\ L(R) &= \{ f(x) \in C(R) : \text{ the coefficient of } x^{2^k} \text{ is zero for all } k>0 \} \\ F(R) &= \{ F(x,y) \in R[[x,y]] : F(x,0) = x,\; F(x,y) = F(y,x),\; F(x,x) = 0,\; F(F(x,y),z) = F(x,F(y,z)) \} \end{align*} The set $G(R)$ is a group under composition, and it acts on $C(R)$ by composition. There is a natural map $C(R)\to F(R)$ that sends a series $f(x)$ to $f^{-1}(f(x) + f(y))$. This is invariant under the action of $G(R)$, so it induces a map $C(R)/G(R)\to F(R)$. Note that $(x+y)^2 = x^2 + y^2 \pmod{2}$, so that series of the form $f(x)=\sum_ka_kx^{2^k}$ (with $a_0=1$) lie in $G(R)$, and one can check that all elements of $G(R)$ have this form. Using this one can show that $G(R)$ acts freely on $C(R)$, that the induced map $C(R)/G(R)\to F(R)$ is a bijection, and that the inclusion $L(R)\to C(R)$ induces a bijection $L(R)\to C(R)/G(R)$.

It turns out that one can define the mod $2$ homology of the object $MO$, and a relatively easy calculation shows that $C(R)$ can be identified with the set of ring maps $H_*MO\to R$. A very general topological construction gives an action of $G(R)$ on $\text{Hom}(H_*MO,R)$, and one can identify this with the action on $C(R)$ mentioned earlier. We now note that this action is free; using this, one can prove that $$ \text{Hom}(\pi_*MO,R) = \text{Hom}(H_*MO,R)/G(R) = L(R). $$ Using this it is easy to recover Thom's description of $\pi_*MO$.

I hope that it is clear even from this very bare outline that there is good reason to study functors from rings to sets such as the functors $C$, $G$, $L$ and $F$ mentioned above. These functors are closely related to schemes in the sense of algebraic geometry. So far we have really only discussed functors that arise from ordinary mod 2 homology. One can obtain more delicate information by using certain generalised cohomology theories. These give functors from suitable categories of spaces to a category of formal schemes. This leads us to many fascinating problems both in topology and in pure algebra (profinite groups, elliptic curves, Galois theory, commutative algebra, class field theory, deformation theory of formal groups, and so on).