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Results tagged with gt.geometric-topology
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user 8032
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).
10
votes
Accepted
The space of contractible loops of a finite dimensional $K(\pi,1)$
The statement is true for a $K(\pi,1)$ but not true for other $X$. Finite dimensionality is not relevant.
Here's a sketch: let $X = K(\pi,1)$ and I will assume $X$ has the homotopy type of a CW com …
- 18.5k
11
votes
Accepted
How does the Framed Function Theorem simplify Cerf Theory?
The framed function theorem tells you that up to "contractible choice" a compact manifold admits a framed function: i.e., a function as you prescribe. Furthermore, a framed function is supposed to giv …
- 18.5k
11
votes
Intuition behind Alexander duality
Yet another way to get it. Alexander duality is closely related to Poincare duality: suppose we can write
$$
S^n = K \cup_A C
$$
where $K$ and $C$ are codimension zero compact manifolds with common bo …
- 18.5k
7
votes
Atiyah duality without reference to an embedding
Assume $M$ is a closed, smooth manifold of dimension $n$. Let $\tau^+$ be the fiberwise one point compactification of its tangent bundle. This is a fiberwise $n$-spherical fibration equipped with a p …
- 18.5k
9
votes
Accepted
Atiyah duality without reference to an embedding
Here is another short construction which is much simpler and just takes a few lines.
Let $M$ be a closed $n$-manifold. Consider the diagonal $M \to M \times M$. It is an embedding. Take its Pontryagi …
- 18.5k
12
votes
The space of framed functions
Eliashberg once told me that the framed function theorem should be a consequence of his work on wrinkled maps.
Igusa and I gave a fairly direct proof that the space of framed functions on the circle …
- 18.5k
11
votes
Accepted
Linking topological spheres
Let $C = S^3 \setminus A$. Alexander duality says that
$$
H_1(C) \cong H^1(A) \cong \Bbb Z\, .
$$
Let $\alpha: S^1 \to C$ be any map representing a generator of $H_1(C)$ (every first homology class …
- 18.5k
3
votes
Accepted
Wall self-intersection invariant for odd-dimensional manifolds?
I doubt that what you are proposing as the receptacle for the obstruction is the correct abelian group. For one thing, you are not taking into account the involution on the canonical double cover of t …
- 18.5k
5
votes
When does an even-dimensional manifold fiber over an odd-dimensional manifold?
No, I don't believe there's a simple solution. But
here's an approach to the problem which indicates how it can be fractured up.
Assume $M,N$ are closed and connected.
If $f\: M \to N$ is homotopic …
- 18.5k
12
votes
1
answer
209
views
A variant of $\ell^2$-cochains
Suppose $X$ is an infinite countable CW complex which satisfies the following property: for all $k$-cells $e$, the number of $(k+1)$-cells incident to $e$ is at most $c_k$, where the latter is some n …
- 18.5k
6
votes
Searching for an unabridged proof of "The Basic Theorem of Morse Theory"
My recollection is that Milnor's proof gives exactly what you are asking. In fact, see the remark on the bottom of page 17 of his book.
- 18.5k
10
votes
Can we decompose Diff(MxN)?
When $N$ is the circle there's a sort of answer. In fact there's a whole chapter in the book
Burghelea, Dan; Lashof, Richard; Rothenberg, Melvin: Groups of automorphisms of manifolds. With an append …
- 18.5k
6
votes
Can one disjoin any submanifold in $\mathbb R^n$ from itself by a $C^{\infty}$-small isotopy?
There are lots of counter-examples.
Here's one:
The fibration $\text{SO}(3) \to \text{SO}(4) \to S^3$ splits, since it is the principal bundle of the tangent bundle of $S^3$, and the latter is paral …
- 18.5k
6
votes
Stratification of smooth maps from R^n to R?
It looks to me that what you are really interested in is the Thom-Boardman stratification of the function space. For that I would recommend the well-written, Stable Mappings and Their
Singularities b …
- 18.5k
11
votes
topological type of smooth manifolds with prescribed homotopy type and pontryagin class
In the $1$-connected case, one may argue as follows:
Let $X$ be a closed $1$-connected smooth $n$-manifold, $n \ge 5$. The theory of the Spivak fibration shows that any homotopy equivalence $f: M^n …
- 18.5k