Kreck's area of specialty is topology, he is particularly interested in differential topology.
He considers himself as a geometer. The central objects of modern geometry are manifolds. These are spaces which locally look like Euclidean spaces. They are models for most physical spaces. For example, the surface of the earth is a 2-dimensional manifold called the 2-sphere. He is mostly interested in the interplay between local and global aspects of manifolds. In geometry, one adds more information to a manifold, namely a metric which allows to speak about the distance between two points. In most of his research the metric does not play a role. But it is almost always in the background.
Professor Kreck is particularly interested in the classification of manifolds. This tries to single out certain characteristics of a manifold, ideally in such a way that they determine the manifold completely. The classification of all manifolds is impossible (in the strict sense it is an unsolvable problem). He always was attracted by classes of manifolds which occur in “daily life”, e.g. in algebraic geometry as solutions of certain polynomial equations, in differential geometry as manifolds with special curvature properties or in physics. The most important tools for classification are homology theories. These are very delicate algebraic measures for the complexity of a space. Apart from using for this the established homology theories, he has constructed various new homology theories.
Recently, he was particularly attracted by a problem which is around for several decades, namely the question whether manifolds admitting a symmetry are less typical than those admitting no symmetry. On the one hand he is interested in constructing examples of asymmetric manifolds and on the other hand he is thinking about a precise formulation of the above vague statement.
Eine Invariante für stabil parallelisierte Mannigfaltigkeiten, Bonner Math. Schriften 66 (1973) (Dissertation)
mit U. Karras, W. Neumann, E. Ossa: Cutting and Pasting of Manifolds; SK-Groups, Math. Lect. Series 1, Publ. or Per. (1973)
mit J. Hermann: Cutting and Pasting of Involutions and Fiberings over the Circle within a Bordism Class, Math. Ann. 214, 11−17 (1975)
Cobordism of odd-dimensional diffeomorphisms, Topology 15, 353−361 (1976)
Bordism of Diffeomorphisms, Bull A.M.S. 82, 759−761 (1976)
Knots and Diffeomorphisms, in: Transformation Groups, London Math. Soc. Lect. Notes Series 26, 295−298 (1977)
Bordismusgruppen von Diffeomorphismen, Habilitationsschrift, Bonn (1977) (ist in erweiterter Form in 10.) erschienen)
Isotopy classes of diffeomorphisms of (k−1)-connected almost parallelizable 2k-manifolds, in: Alg. Top. Aarhus 1978, Springer Lect. Notes 763, 219−232 (1979)
Manifolds with unique differentiable structure, Topology 23, 219−232 (1984)
Bordism of diffeomorphisms and related topics, Springer Lect. Notes 1069 (1984)
mit J. Schafer: Classification versus stable classification of manifolds: some examples, Comm. Math. Helv. 59, 12−38 (1984)
Some closed 4-manifolds with exotic smooth structure, in Proc. Aarhus Symposium 1982, Springer Lect. Notes 1051, 246−262 (1984)
Exotische Strukturen auf 4-Mannigfaltigkeiten, Jber. DMV 88, 124−145 (1986)
Smooth structures on closed 4-manifolds up to connected sum with (S2×S2)’s, preprint (1984)
An extension of results of Browder, Novikov and Wall, preprint (1985)
mit S. Finashin und O. Viro: Exotic embeddings of surfaces in the 4-sphere, Bull A.M.S. 17, 287−290 (1987)
Solution of Problem 8 (R. Thom) in F. Hirzebruch’s problem list from Annals of Math. 1954, erschienen in Hirzebruch’s collected works, Band 1, 771−772 (1987)
mit I. Hambleton: On the classification of topological 4-manifolds with finite fundamental group, Math. Ann. 280, 85−104 (1988)
mit S. Stolz: A diffeomorphism classification of 7-dimensional homogeneous Einstein manifolds with SU(3)×SU(2)×U(1)-symmetry, Annals of Mathematics 127, 373−388 (1988)
Ist das Universum exotisch?, Populärwiss. Artikel, in engl. und span. Übersetzung erschienen in Universitas, 1988
mit I. Hambleton: Smooth structures on algebraic surfaces with cyclic fundamental group, Inventiones Math. 91, 53−59 (1988)
mit S. Finashin und O. Viro: Non-diffeomorphic but homeomorphic knottings of surfaces in the 4-sphere, Springer Lect. Notes 1346, 157−198 (1988)
mit I. Hambleton: Smooth structures on algebraic surfaces with finite fundamental group, Inventiones Math. 102, 109−114 (1990)
mit Th. Fischer: Rohlin invariants, integrality of some spectral invariants and bordism of diffeomorphisms, Topol. and its Appl. 37, 181−199 (1990)
On the homeomorphism classification of smooth knotted surfaces in the 4-sphere, in Proc. Geometry of low-dimensional manifolds, London M.S. Lect. Notes Series 150, 63−72 (1990)
mit G. Triantafillou: On the classification of manifolds up to finite ambiguity, Can. Jour. of Math. 43, 356−370 (1991)
mit S. Stolz: Some nondiffeomorphic homeomorphic homogeneous 7-manifolds with positive sectional curvature, Journal Diff. Geo. 33, 465−486 (1991)
Positive Krümmung und Toplogie, Jahresber. DMV, Jubiläumstagung 1990, 327−341 (1992)
mit I. Hambleton: Cancellation of lattices and finite two-complexes, J. reine angew. Math. 442, 91−109 (1993)
mit I. Hambleton: Cancellation of hyperbolic forms and topological 4-manifolds, J. reine angw. Math. 443, 21−47 (1993)
mit I. Hambleton: Cancellation, elliptic surfaces and the topology of certain 4-manifolds, J. reine angw. Math. 444, 79−100 (1993)
mit S. Stolz: Nonconnected moduli spaces of positive sectional curvature metrics, Journal AMS, 6, 825−850 (1993)
mit S. Stolz: HP2-bundles and elliptic homology, Acta Math. 171, 231−261 (1993)
mit I. Hambleton: Cancellation results for 2-complexes and 4-manifolds and some applications, London Math. Soc. Lect. Note Series, 281−308
mit I. Hambleton und P. Teichner: Nonorientable 4-manifolds with fundamental group of order 2, Trans. AMS, 344, 649−665 (1994)
Positive Krümmung und Topologie, Nordrhein-Westf. Akademie d. Wissenschaften 404 (1994)
Ist 4 denn noch normal?, Jubiläumsband Oberwolfach, Springer Verlag (1994)
mit W. Lück und P. Teichner: Counterexamples to the Kneser conjecture in dimension 4, Comm. Math. Helv. 70, 423−433 (1995)
mit W. Lück und P. Teichner: Stable Prime Decomposition of four-manifolds, Ann. Math. Stud. 138, 251−269 (1995)
Surgery and Duality, Ann. Math. (2) 149, No.3, 707−754 (1999).
A connection between Kaluza-Klein and Calabi-Yau via topological resolution of singularities, preprint (1999) (will be incorporated into 50.))
A Guide to the Classification of Manifolds, Ann. Math. Stud. 145, 121−134 (2000).
Smooth h-cobordisms between 1-connected 4-manifolds, Geometry and Topology, Vol. 5, 1−6 (2001)
An inverse to the Poincare conjecture, Archiv der Math. 77 (2001), 98−106
mit W. Lück and E. Leichtnam: On the cut and paste property of higher signatures of a closed oriented manifold, Topology 41, 725−744 (2002)
mit S. Klaus: A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 3, 617−623
Differential Algebraic Topology, Monographie, preprint (2004)
mit M. Katz and A. Suciu: Free abelian covers, short loops, stable length, and systolic inequalities, preprint (2002), accepted by Math. Ann.
mit I. Hambleton: Self equivalences of 4-manifolds, Math. Z. 248 (2004), No. 147−172.
Stratifolds, Monographie, in preparation
mit W. Singhof: Uniqueness of homology and cohomology on manifolds, preprint (2004)
Simply connected asymmetric manifolds, preprint, Heidelberg 2004
mit A. Skopenkov: Inertia groups of smooth embeddings, preprint, 2004 submitted to Geometry and Topology
mit W. Lück: The Novikov Conjecture, Geometry and Algebra, Monographie, Birkhäuser