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Results tagged with gt.geometric-topology
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user 6666
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
33
votes
Accepted
"Largest" finite-dimensional Lie subgroups of Diff(S^n), are they known?
You can make big Lie groups act effectively on small manifolds by cheating: make the group a product of groups, with each factor acting by compactly supported diffeomorphisms on a different disjoint o …
- 55.9k
31
votes
Examples of non-diffeomorphic smooth manifolds with diffeomorphic tangent bundle
Another good $2$-dimensional example is torus punctured once and sphere punctured three times. These become diffeomorphic when crossed with $\mathbb R$, and they have trivial tangent bundles. More gen …
- 55.9k
26
votes
Accepted
Extending a diffeomorphism of the sphere $S^2$ to the ball $D^3$
More a survey of related things than an answer, but here goes.
Let's write $D(n)$ for the space of compactly supported diffeomorphisms $\mathbb R^n\to \mathbb R^n$. A reasonable guess might be that t …
- 55.9k
26
votes
Accepted
"Affine communication" for topological manifolds
There are piecewise linear counterexamples in dimension $2$.
Arrange $2m$ evenly spaced rays $R_i$ around the origin, $m\ge 3$. If $C$ is a convex neighborhood of the origin, let $r_i$ be the recipr …
- 55.9k
22
votes
Fixed-point free diffeomorphisms of surfaces fixing no homology classes
Here's an example with $g=2$. Let $T$ be the torus $\mathbb C/L$, where $L$ is the lattice spanned by $1$ and $\zeta=e^{2\pi i/6}$. Let $f:T\to T$ be induced by multiplication by $\zeta$. This is a di …
- 55.9k
17
votes
Accepted
Can a cyclic group of prime order act on a rationally acyclic finite dimensional complex and...
Yes, I think you can make an example like this (for $p=2$, but it generalizes).
Let $R$ be the group ring $\mathbb Z[C_2]=\mathbb Z[x]/(x^2-1)$. Make a chain complex of free $R$-modules
$$
M_0 \leftar …
- 55.9k
15
votes
Accepted
Simply connected slices
Yes, I think so. Let's show that every compact set $K\subset \Omega$ is contained in some compact contractible subset of $\Omega$. We use the fact that in a simply connected open subset of the plane e …
- 55.9k
15
votes
Accepted
Smooth structures compatible with a given C^1 structures
See my answer to this question.
Consider the space (in some appropriate sense ) of all invertible germs of $C^{\infty}$ maps from $\mathbb R^n$ to itself fixing the origin. This is homotopy equivale …
- 55.9k
14
votes
Associativity of topological join and join of spheres
If $A$ and $B$ are subsets of $\mathbb R^n$ then you can map $A\times I\times B$ to $\mathbb R^n$ by $(a,t,b)\mapsto (1-t)a+tb$. If the resulting continuous map $A*B\to \mathbb R^n$ happens to be one …
- 55.9k
14
votes
Accepted
Intersection product in a manifold, taking values in one factor
I don't think you have to get involved with strata or transversality at all. Poincare duality (Edit: and cap products) will do all the work. Let's assume:
$M$ is a compact oriented $n$-manifold.
$\ …
- 55.9k
13
votes
Accepted
What is the status of the PL-pseudoisotopy stability theorem?
As you say, Hatcher once argued that the map $\sigma_M^{PL}:C^{PL}(M)\to C^{PL}(M\times I)$ is $k$-connected where $k$ is roughly $n/3$, but the proof was not all there.
And as you say Igusa later pr …
- 55.9k
13
votes
Homotopy groups of spaces of embeddings
Suppose that $M$ is compact, $N$ is simply connected and has finitely generated homology, and the codimension $n-m$ is at least $3$. Then the space $Emb(M,N)$ is such that
(1) for every basepoint $\p …
- 55.9k
12
votes
Accepted
Embedding the product of three circles in the 4-sphere.
No. Suppose that the rank of $H^1(V_1)$ is zero, so that the rank of $H^1(V_2)$ is three and (by looking at the Mayer-Vietoris sequence again) the rank of $H^2(V_2)$ is zero. Take two independent elem …
- 55.9k
12
votes
Accepted
Any map of a contractible complex to itself has a fixed point
You have to assume that your complex is finite. Then the Lefschetz Fixed Point Theorem definitely says that $f$ must have a fixed point if the (homologically defined) Lefschetz number of $f$ is not ze …
- 55.9k
12
votes
Accepted
Simplicial replacements in smoothing theory
In the topological category the usual compact-open topology does the job. (EDIT: Or rather I suppose you might have to modify it a little so that $h\mapsto h^{-1}$ is continuous.) At times you might w …
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