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

- My papers on MathSciNet. (This will only work if you are at a university that subscribes to MathSciNet.)
- My papers on the ArXiv
- My PhD thesis
- Some additional preprints, mostly in a half-baked state.
- Projects that I am interested in. Some of these are suitable for PhD students.
- Mathematica representation of the unstable homotopy groups of spheres

## 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.- 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.)
- 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 Emb(M,N) of embeddings, and our task is to understand the set π_{0}Emb(M,N) of path-components in this space. One approach is to write M as a union of simpler manifolds, say M_{1},..,M_{k}. One can then use the methods of homotopy theory to relate π_{0}Emb(M,N) to the higher homotopy groups of Emb(M_{i},N), and various related spaces. - 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 x I, we see that MO_{*}is actually an algebra over 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 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. - 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 + Q' + F is isomorphic to P' + Q + F for some finitely generated projective module F. Here + denotes the direct sum. 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 x 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) x K_{j}(R) -> K_{i+j}(R) for i+j <= 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 π_{i}K(R) = K_{i}(R) for i <= 2, and then to define K_{i}(R) to be π_{i}K(R) for i>2. This idea was very successful, and the resulting theory has had many applications in number theory and so on. - 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 π
_{1}BG = G and π_{k}BG = 0 for k != 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 the reals, and let B be the set of finite subsets A < 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.

_{*}, as mentioned above. Let R be an algebra over Z/2. We need to consider various sets associated to R.

C(R) = { formal power series f(x) over R such that f(0)=0 and f'(0) = 1 }

G(R) = { formal power series f(x) over R such that f(0)=0 and f'(0) = 1 and f(x+y) = f(x) + f(y) }

L(R) = { formal power series f(x) in C(R) such that the
coefficient of x^{2k} is zero for all k>0 }

F(R) = { formal power series F(x,y) over R such that F(x,0) = x, F(x,y) = F(y,x), F(x,x) = 0, F(F(x,y),z) = F(x,F(y,z)) }

The set G(R) is a group under composition, and it acts on C(R) by
composition. There is a natural map C(R) -> 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) -> F(R). Note that
(x+y)^{2} = x^{2} + y^{2} mod 2, so that
series of the form

S_{k}a_{k} x^{2k}

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) -> F(R) is a bijection, and that the inclusion L(R) -> C(R) induces a bijection L(R) -> 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 -> R. A very general
topological construction gives an action of G(R) on
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

Hom(π_{*}MO,R) =
Hom(H_{*}MO,R)/G(R) = L(R).

Using this it is easy to recover Thom's description of π_{*}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).