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Results tagged with gt.geometric-topology
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user 290
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).
25
votes
Accepted
What is the Euler characteristic of a mapping space?
This Euler characteristic usually won't be well-defined. For example, take $A = S^1$ and $B = S^3$. Then the mapping space $[A, B]$ is the free loop space $L S^3$, which decomposes as a product
$$L S …
- 118k
16
votes
Accepted
quotient space of Eilenberg-MacLane space
Suppose a group $H$, not necessarily finite, acts on an Eilenberg-MacLane space $BN$. The homotopy quotient $BN/H$ (which agrees with the ordinary quotient if the action of $H$ is free) fits into a fi …
- 118k
16
votes
Accepted
Homotopy groups other than $\pi_1$ : what are they good for?
Here is a simple example: even if all you care about is computing $\pi_1$, sometimes the easiest way to do it is to use the long exact sequence of a fibration, which requires knowing something about a …
14
votes
Accepted
What is the max number of points in R^3, interconnected by generic curves?
Take straight lines connecting the points $(t, t^2, t^3), t \in \mathbb{N}$. As far as I can tell you can also boost this to $t \in \mathbb{R}$. The point here is that two distinct lines between poi …
- 118k
14
votes
Applications of knot theory
Colin Adams' The knot book discusses the following applications:
Knotting in DNA,
Molecular knots,
Statistical mechanics (e.g. the Potts model).
Constructing invariants of knots is also related t …
13
votes
Computation on characteristic classes
I wrote a blog post that turned into quite a nice exercise in characteristic classes. The goal was to compute the cohomology of a smooth hypersurface of degree $d$ in $\mathbb{CP}^3$, as a ring. This …
- 118k
12
votes
Accepted
classifying space of orthogonal groups
$BO$ is the connected component of the zeroth space of a spectrum called the real K-theory spectrum. This spectrum represents a cohomology theory, namely real K-theory, and this means that $BO$ has mu …
- 118k
10
votes
Accepted
Are there compact flat fiber bundles with "truly" infinite structure group?
Any finitely presented group occurs as the fundamental group of a smooth compact manifold, so the question reduces to whether we can find a finitely presented group $\pi$ acting on a smooth compact ma …
- 118k
8
votes
1
answer
499
views
Reference request: the "Kauffman bracket skein category"?
There should be a category $3\text{CobTang}$ whose
objects are some kind of surfaces with a finite set of marked points
morphisms $M : S \to T$ are some kind of $3$-dimensional cobordisms …
- 118k
6
votes
Tangent bundle of smooth closed simply-connected $4$-manifold $w_1 = w_2 = 0$ can be trivial...
Yes. For manifolds of dimension $\le 7$ the obstructions to trivializing any real vector bundle are $w_1, w_2$, and a class $\frac{p_1}{2} \in H^4(BSpin(n), \mathbb{Z})$ ($n \ge 3$), the fractional fi …
- 118k
5
votes
Accepted
A Question about SO(n)
There are a pair of double covers $\text{SU}(2) \times \text{SU}(2) \to \text{SO}(4) \to \text{SO}(3) \times \text{SO}(3)$, and the first resp. the second more or less reduces the classification of fi …
- 118k
5
votes
A question about axes of symmetry in the plane.
Yes. Suppose without loss of generality that the two lines are the x- and y-axes. Some point of J lies in some quadrant, so corresponding points a, b, c, d of J exist in every quadrant by symmetry. …
- 118k
3
votes
Distinct manifolds with the same configuration spaces?
Let $M = \mathbb{R}^2, N = D^2$. In both cases I think the configuration space of $k$ distinct unordered points has the homotopy type of $K(B_k, 1)$. More generally I think we can take $N$ to be a man …
- 118k
2
votes
"C choose k" where C is topological space
The papers I was thinking of are actually by Propp:
Euler measure as generalized cardinality, and
Exponentiation and Euler measure
(although I think these papers are outdated and more is known these …
- 118k
2
votes
Knot theory without planar diagrams?
As far as quantum invariants, there is Witten's Quantum Field Theory and the Jones Polynomial, which gives a 3-dimensional definition of the Jones polynomial (usually defined, as you say, using 2-dime …
- 118k