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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).
14
votes
How to prove the connected sum of two closed aspherical n-manfolds (n >2) is not asperical?
Here's a different way to see it. Let $M$ and $N$ be aspherical of dimension at least 3.
Then the wedge $M \vee N$ is aspherical (but not a manifold). Let $M\sharp N$ be the connected sum. Then we get …
- 18.5k
5
votes
Homotopy type of the plane minus a sequence with no limit points
Here is a more naive solution, as least if the sequence is countable. Let
$\Bbb N \subset \Bbb R^2$ be the embedding defined by the sequence.
Then there is an isotopy from this embedding to the stand …
- 18.5k
7
votes
Manifolds covered by a single disc
The answer (to the first question) is yes in the smooth case: If $M^m$ is closed and compact, then there is a Morse-Smale function on $M$ with a single critical point of index $m$. Work by Lizhen Qin …
- 18.5k
1
vote
Regular homotopy invariance of Wall's self-intersection form.
Even more is true in the context of surgery theory: let
$q \ge 3$ with $q$ odd, assume $M$ is a $1$-connected closed smooth manifold of dimension $2q$. Let $$I^{\text{fr}}_q(M)$$ denote the space of i …
- 18.5k
8
votes
What characteristic class information comes from the 2-torsion of $H^*(BSO(n);Z)$?
See
Brown, Edward, The cohomology of $B\text{SO}_n$ and $B\text{O}_n$ with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288
for an answer to your questions. (As for applications: …
- 18.5k
12
votes
Accepted
"C choose k" where C is topological space
I presume you are asking is whether one can make sense of $\binom{X}{k}$ as a space in such a way that it relates to the formula you are asking about.
The answer is yes.
First let $l = 1$. For a sp …
- 18.5k
17
votes
Accepted
Immersions of surfaces in $\mathbb{R}^3$
(Based on the comment of Mariano Suárez-Álvarez, there was a false assumption in my original answer. This is an attempt to correct it.)
1) Let $M$ be a closed smooth manifold with $k < n$. According …
- 18.5k
5
votes
Accepted
Is the Artin Spin construction related to the suspension functor?
This question is answered in section 4 of my first paper (with Alex Suciu)
Klein, John R.; Suciu, Alexander I.
Inequivalent fibred knots whose homotopy Seifert pairings are isometric.
Math. Ann. 289 ( …
- 18.5k
9
votes
Compelling evidence that two basepoints are better than one
I just wanted to add something to the discussion about the utility of adding additional basepoints. It turns out this is crucial for understanding
certain aspects of embedding theory. See the bottom o …
- 18.5k
0
votes
First usage of the terms pseudo-isotopy and concordance in manifold theory
MathSciNet refers to a paper of Bing from 1959 with the term "pseudo-isotopy" in the math review:
Bing, R. H.
The cartesian product of a certain nonmanifold and a line is
E^4. Ann. of Math. (2)70(1959 …
- 18.5k
18
votes
What is the intuition behind the Freudenthal suspension theorem?
The Freudenthal theorem is really a special case of the phenomenon called "homotopy excision" aka the Blakers-Massey triad theorem. The idea is that one has an inclusion
$$
(C_-X,X) \to (\Sigma X,C_+X …
- 18.5k
2
votes
Has this kind of question in topology a special name?
How about $\pi_0(\text{Homeo}(X))$?
The papers of Weiss and Williams (automorphisms of manifolds and algebraic $K$-theory...) are relevant since they reduce computations of $\pi_i(\text{aut}(X))$ for …
- 18.5k
5
votes
Accepted
When does an antipodal map on a manifold extend to the antipodal map on a spheres
Let me elaborate on my comment above. Suppose $M$ is a manifold equipped with a smooth $\Bbb Z_2$ action that is also free. Then there is an equivariant smooth embedding $M \to S^j$, for some $j$, whe …
- 18.5k
0
votes
Poset fiber theorems under a special assumption on the poset map?!
In the following, I shall assume that the posets $P$ and $Q$ are finite.
Then it is at least true from the condition that $f^{-1}(q)$ is contractible for all $q \in Q$ that the map $|f|$ is a homolog …
- 18.5k
2
votes
What should be taught in a 1st course on smooth manifolds?
I do have one addition to make to the above. At our university we usually use a combination of Guillemin and Pollack and Milnor. There is another approach at a first course which some have found usefu …