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Results tagged with gt.geometric-topology
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user 8103
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).
29
votes
In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?
The higher homotopy groups of the complement of a knot in $S^3$ are all trivial. This is not true for a knot in $\mathbb{R}^3$ (consider a $2$-sphere enclosing the image of the knot).
- 35.9k
27
votes
Accepted
Can one deform an immersion of a 3-manifold in $\mathbb R^4$ to an embedding in $\mathbb R^6$?
Quoting Theorem F of this paper by Ulrich Koschorke:
For any self-transverse immersion $j$ of a closed 3-manifold $M$ into $\mathbb{R}^4$ the following integers are equal modulo 2:
the Eul …
- 35.9k
22
votes
Is there a $4$-manifold which Immerses in $\mathbb{R}^6$ but doesn't Embed in $\mathbb{R}^7$?
Edit: The answer below is incorrect. In fact, $\bar{w}_3(\mathbb{R}P^2\times\mathbb{R}P^2)=0$ (thanks to Rafal Walczak for pointing this out) so by the cited result $\mathbb{R}P^2\times\mathbb{R}P^2$ …
- 35.9k
21
votes
Accepted
Link such that deleting any two components leaves an unlink
Yes, this is done in
Penney, D.E., Generalized Brunnian links, Duke Math. J. 36, 31-32 (1969). ZBL0176.22201.
Call a link $(n,k)$-Brunnian if it has $n$ components, and every sublink with $m$ compo …
- 35.9k
21
votes
Accepted
Does every dimension $n\geq4$ admit a manifold with an exotic smooth structure?
Yes. For every $n\ge 5$ there are exotic tori.
In fact, the PL-structures on $T^n$ are in one-to-one correspondence with $H^3(T^n;\mathbb{Z}/2)$, and every one of these is smoothable (Reference: "Su …
- 35.9k
18
votes
Accepted
Wu formula for manifolds with boundary
A relative Wu formula for manifolds with boundary is discussed in Section 7 of
Kervaire, Michel A., Relative characteristic classes, Am. J. Math. 79, 517-558 (1957). ZBL0173.51201.
In particular, t …
- 35.9k
16
votes
Can the nth projective space be covered by n charts?
It seems worth giving the cup-length argument, as it's relatively short and sweet.
Suppose $\mathbb{R}P^n=U_1\cup\cdots\cup U_n$, with each $U_i\approx\mathbb{R}^n$, and let $c\in H^1(\mathbb{R}P^n;\m …
- 35.9k
15
votes
Accepted
What is the 'non-intuitive' part in sphere eversion (turning inside out)?
Watch Outside In (something we should all do anyway, to commemorate Bill Thurston's passing).
To understand the mathematics behind sphere eversions, you should first get a good intuition for the conc …
- 35.9k
14
votes
Accepted
Homology of infinite loop spaces $QX$
Let $X$ be a connected space, and let $\lbrace x_\lambda\rbrace$ be a homogeneous basis for $H_\ast(X;\mathbb{F}_2)$. Then
$$H_\ast(QX;\mathbb{F}_2) = \mathbb{F}_2 [Q^I x_{\lambda} \mid I\mbox{ admis …
- 35.9k
14
votes
Accepted
Twisting bordism classes
The correct definition of bordism should have this built in. More precisely, two singular manifolds $(M_0^n,f_0)$ and $(M_1^n,f_1)$ in a space $X$ are oriented bordant if there exists a smooth manifol …
- 35.9k
14
votes
Vector bundle for prescribed Stiefel-Whitney classes
In addition to Wu's formula mentioned in Oscar's answer, there are further obstructions coming from the fact that the mod 2 reduction of $p_i$ (the $i$th Pontryagin class) is $w_{2i}^2$. So your class …
- 35.9k
14
votes
Integral homology classes that can be represented by immersed submanifolds but not embedded ...
This is a great question, and I don't have an answer but this is too long for a comment.
Working mod $2$, a codimension $k$ homology class $z\in H_{m-k}(M;\mathbb{Z}/2)$ is realizable by an embedding …
- 35.9k
13
votes
Where is the Steenrod Realization problem at?
A lot depends on what you want to know about realizability. You could argue that Thom's paper settles the problem: in the mod 2 case, every homology class is realizable by a map; in the integral case, …
- 35.9k
12
votes
Isotopy extension theorems
The original reference for the topological isotopy extension theorem is Corollary 1.4 of
Edwards, Robert D.; Kirby, Robion C.
Deformations of spaces of imbeddings.
Ann. Math. (2) 93 1971 63–88.
No …
- 35.9k
11
votes
Can a Morse function define a unique structure on a closed manifold?
In looking for counter-examples it helps to notice that $M$ and $N$ have CW structures with the same numbers of $i$ cells, and hence they have the same Euler characteristic.
It turns out that there i …
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