6,345 research outputs found
The Loop Space Homotopy Type of Simply-connected Four-manifolds and their Generalizations
We determine loop space decompositions of simply-connected four-manifolds,
-connected -dimensional manifolds provided , and
connected sums of products of two spheres. These are obtained as special cases
of a more general loop space decomposition of certain torsion-free
-complexes with well-behaved skeleta and some Poincar\'{e} duality
features.Comment: Adv. Math., to be publishe
Involutions on S^6 with 3-dimensional fixed point set
In this article, we classify all involutions on S^6 with 3-dimensional fixed
point set. In particular, we discuss the relation between the classification of
involutions with fixed point set a knotted 3-sphere and the classification of
free involutions on homotopy CP^3's.Comment: 23 page
Orientation reversal of manifolds
We call a closed, connected, orientable manifold in one of the categories
TOP, PL or DIFF chiral if it does not admit an orientation-reversing
automorphism and amphicheiral otherwise. Moreover, we call a manifold strongly
chiral if it does not admit a self-map of degree -1. We prove that there are
strongly chiral, smooth manifolds in every oriented bordism class in every
dimension greater than two. We also produce simply-connected, strongly chiral
manifolds in every dimension greater than six. For every positive integer k, we
exhibit lens spaces with an orientation-reversing self-diffeomorphism of order
2^k but no self-map of degree -1 of smaller order.Comment: This is the update to the final version. 22 page
Classification of smooth embeddings of 4-manifolds in 7-space, I
We work in the smooth category. Let N be a closed connected n-manifold and
assume that m>n+2. Denote by E^m(N) the set of embeddings N -> R^m up to
isotopy. The group E^m(S^n) acts on E^m(N) by embedded connected sum of a
manifold and a sphere. If E^m(S^n) is non-zero (which often happens for
2m<3n+4) then no results on this action and no complete description of E^m(N)
were known. Our main results are examples of the triviality and the
effectiveness of this action, and a complete isotopy classification of
embeddings into R^7 for certain 4-manifolds N. The proofs are based on the
Kreck modification of surgery theory and on construction of a new embedding
invariant.
Corollary. (a) There is a unique embedding CP^2 -> R^7 up to isoposition.
(b) For each embedding f : CP^2 -> R^7 and each non-trivial knot g : S^4 ->
R^7 the embedding f#g is isotopic to f.Comment: 22 pages, no figures, statement and proof of the Effectiveness
Theorem correcte
One-connectivity and finiteness of Hamiltonian -manifolds with minimal fixed sets
Let the circle act effectively in a Hamiltonian fashion on a compact
symplectic manifold . Assume that the fixed point set
has exactly two components, and , and that . We first show that , and are simply connected. Then we
show that, up to -equivariant diffeomorphism, there are finitely many such
manifolds in each dimension. Moreover, we show that in low dimensions, the
manifold is unique in a certain category. We use techniques from both areas of
symplectic geometry and geometric topology
A classification of smooth embeddings of 3-manifolds in 6-space
We work in the smooth category. If there are knotted embeddings S^n\to R^m,
which often happens for 2m<3n+4, then no concrete complete description of
embeddings of n-manifolds into R^m up to isotopy was known, except for disjoint
unions of spheres. Let N be a closed connected orientable 3-manifold. Our main
result is the following description of the set Emb^6(N) of embeddings N\to R^6
up to isotopy.
The Whitney invariant W : Emb^6(N) \to H_1(N;Z) is surjective. For each u \in
H_1(N;Z) the Kreck invariant \eta_u : W^{-1}u \to Z_{d(u)} is bijective, where
d(u) is the divisibility of the projection of u to the free part of H_1(N;Z).
The group Emb^6(S^3) is isomorphic to Z (Haefliger). This group acts on
Emb^6(N) by embedded connected sum. It was proved that the orbit space of this
action maps under W bijectively to H_1(N;Z) (by Vrabec and Haefliger's
smoothing theory). The new part of our classification result is determination
of the orbits of the action. E. g. for N=RP^3 the action is free, while for
N=S^1\times S^2 we construct explicitly an embedding f : N \to R^6 such that
for each knot l:S^3\to R^6 the embedding f#l is isotopic to f.
Our proof uses new approaches involving the Kreck modified surgery theory or
the Boechat-Haefliger formula for smoothing obstruction.Comment: 32 pages, a link to http://www.springerlink.com added, to appear in
Math. Zei
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