Peter Eccles's Home Page
Home Page for Peter Eccles
My main research interest is in the application of algebraic topology
to differential topology. Its main focus in recent years has been the
study of multiple point sets of immersions of smooth manifolds. If
the immersion is in `general position' then the multiple point
sets will themselves be the the images of immersions of other immersed
manifolds. We can ask about the properties of these multiple point
manifolds, which manifolds can arise and to what extent the original
manifold determines the multliple point manifolds.
An example of a result in this area is the one I started with. For a
codimension 1 immersion of a closed n dimensional manifold in
Euclidean (n+1)-space the set of (n+1) fold points (points in the
image of (n+1) distinct points in the manifold) is a discrete set. It
turns out that if the manifold is orientable then the number of (n+1)
fold points can be odd if and only if n = 1 or n = 3. In these cases
we can take the manifold to be the n-sphere. This result was obtained
by translating the problem into a problem in algebraic topology by
means of cobordism theory and the homotopy theory of Thom complexes.
Other problems of the same type can be explored by the same means and
tackling these brings into play a whole range of techniques from
The same question for non-orientable manifolds
is closely connected to classical problems in the
homotopy groups of spheres: the Hopf invariant problem and the
Kervaire invariant problem.
Double point manifolds of immersions of spheres in Euclidean
space, Proceedings of the William Browder Sixtieth Birthday
Conference, March 1994, Annals of Math. Studies, 138, Princeton
University Press, 1997, 125-137.
(with Paul R Turner and W Stephen Wilson) On the Hopf ring for the
sphere, Math. Z. 224 (1997), 229-233.
(with Pyotr M Akhemt'ev) A geometrical proof of Browder's result
on the vanishing of the Kervaire invariant, (in Russian)
Tr. Math. Inst. Steklova 225 (1999), 46-51,
(in English) Proc. Steklov
Inst. Math. 225 (1999), 40-44.
(with Mohammad A Asadi-Golmankhaneh) Determining the
characteristic numbers of self-intersection manifolds, J. London
Math. Soc. (2) 62 (2000), 278-290.
(with Mohammad A Asadi-Golmankhaneh)
Double point self-intersection surfaces of immersions,
Geometry and Topology 4 (2000), 149-170.
(with Mark Grant)
Bordism classes represented by multiple point manifolds of immersed manifolds,
Proceedings of the Keldysh Centenary Conference, Moscow, August 2004.
(with Mark Grant)
Bordism groups of immerersions and classes represented by self-intersections,
Algebr. Geom. Topol., 7 (2007) 1081-2097.
files on arXiv.org
(with Pyotr Akhmet'ev)
The relationship between framed bordism and skew-framed bordism,
(with Mark Grant)
Self-intersections of immersions and Steenrod operations,
Acta Math. Hungar. 137 (4) 2012, 272-281.
Introduction to Mathematical Reasoning: numbers, sets and functions,
Cambridge University Press, December 1997, reprinted 1998, 2000(twice), 2001, 2002, 2004.
For further information and ordering details see
Great Britain or
United States of America.
Book (not mathematics):
The Presence in the Midst: reflections on discernment (Swarthmore Lecture 2009), Quaker Books, London, 2009.
Journal of Homotopy and Related Structures.
Graduate students past and present:
Sadoon M Al-Musawe, Multiplications on certain H-spaces, MSc 1974
Salih Karadag, Intersection points of immersed manifolds,
Khalid Felali, Intersection points of immersed manifolds,
William P R Mitchell, An application of and the derivation of a
certain splitting, MSc 1984
William P R Mitchell, p-fold intersection points and their
relation with MU(n), PhD 1986
Richard J Morris, Immersions of manifolds, MSc 1988
Adam J Harrison, Self-intersection points of immersed
manifolds, MSc 1990
Paul R Turner, A filtration of the homotopy groups of
spheres, MSc 1991
Paul R Turner, On the homology of certain infinite loop
spaces, PhD 1993
Mohammad A Asadi-Golmankhaneh, Self-intersection manifolds of
immersions, PhD 1998
Christopher I Hazell, Immersions up to bordism, PhD 1998
Jonathan M A Burgess, On self-intersections and the stable homotopy of D2MO(2), PhD 2002
Mark Grant, Bordism of immersions, PhD 2006
Hadi Zare, On spherical classes in H*QSn, PhD 2009
Azzah Al-Shehry, The double point surfaces of immersions in complex projective spaces, PhD 2010
Alex Longdon, current student.
MATH31051/MATH41051/61051 Introduction to topology: Course unit web page
MATH20132 Calculus of several variables: Course unit web page
MATH31072/MATH41072/MATH61072 Algebraic topology: Course unit web page
Students are welcome to call on me in my office (1.111 in the
Alan Turing Building) at any time. If I am busy I will
ask you to come back another time and if possible arrange one. If I
am not in my office then leave a note under my door or send me an
email saying when you will return. I
will try and make a point of being in my office available to students
on Tuesday (1.00 - 2.00) and Thursday (4.00 - 5.00).
However, sometimes I will have other commitments at these times.
Last revised 25 September 2013