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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).
3
votes
Extending homotopy equivalence between manifolds with boundary
Tangent Stiefel-Whitney classes provide some obstructions. If $\bar f:M_1\to M_2$ is a homotopy equivalence, then $\bar f^\ast w_j(M_2)=w_j(M_1)$, and if $\bar f$ extends a map $f:N_1\to N_2$ where $N …
- 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
7
votes
index of morse functions and homotopical dimension
Suppose that $M$ has a proper Morse function $f\ge 0$ with all critical points of index at most $k$. Then homotopically $M$ can be made of low-dimensional cells: it has homotopical dimension at most $ …
- 55.9k
3
votes
One question on cup product and torsion elements
You can also make an example involving a closed surface. Let $X$ be the connected sum of a torus $T$ and a projective plane, and let $f:X\to T$ be nontrivial on $H^2$. Two elements of $H^1(T)$ whose c …
- 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
10
votes
Is the topological concept of collapsible useful?
To add a little to Ryan's answer:
This topic is maybe not exactly a part of algebraic topology. It's more like an area of application of algebraic topology to certain important special classes of sp …
- 55.9k
8
votes
Accepted
Conventions for definitions of the cap product
This is just an expanded version of Tyler's comment, I think.
Let's use a, b, c for cochains, x, y, z for chains, [a,x] for the value of a cochain on a chain. I'll be lazy and write $ab$ for $a\cup b …
- 55.9k
5
votes
Accepted
Local homology of degenerate critical points
$(x^2+y^2)z^2-c(x^2+y^2+z^2)^2$ for small positive $c$.
More generally $f-cr^{2d}$ where $f\ge 0$ is a homogeneous polynomial function of degree $2d$ in $n$ variables. The local homology at the orig …
- 55.9k
7
votes
Are there oriented $4k+2$ manifolds such that $im(H_{2k+1}(M; Z/2)\to H_{2k+1}(M, \partial M...
Martin O's answer is very nice. So in an oriented $2n$-manifold with $n$ odd the mod $2$ self-intersection of any $n$-dimensional mod $2$ homology class is $0$.
Looking for a more geometric explanat …
- 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
8
votes
Accepted
Finite-dimensional subgroups of circle diffeomorphism group
If $G$ is a connected Lie group acting transitively and faithfully on a connected smooth $1$-manifold, then $G$ is at most $3$-dimensional; in fact its Lie algebra embeds in that of $SL_2(\mathbb R)$. …
- 55.9k
9
votes
Accepted
Generalised linking numbers (where they shouldn't be)
Sort of obvious, but: in general you get lots of linking numbers. If spaces $K$ and $L$ are mapped disjointly into $\mathbb R^n$ then for every $a\in H_i(K)$ and $b\in H_j(L)$ with $i+j=n-1$ you get a …
- 55.9k
2
votes
What is the normalizer of the circle in the diffeomorphism group of the 2-sphere?
Let's call your coordinates $\phi$ and $\theta$, as is more usual. Thus $(x,y,z)=(sin \phi\ cos\theta, sin\phi\ sin\theta, cos\phi)$.
Yes, in the centralizer of $SO(2)$ there is the group that leave …
- 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
1
vote
Is the intersection of boundaries of convex bodies a topological sphere?
Here is a well known counterexample.
Edit: Oh, sorry, a disjoint union of spheres is allowed. I misread the question.
- 55.9k