# Questions tagged [gn.general-topology]

Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

**22**

votes

**6**answers

2k views

### Is there a topological description of combinatorial Euler characteristic?

There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and ...

**49**

votes

**16**answers

10k views

### Atiyah-Singer index theorem

Every year or so I make an attempt to "really" learn the Atiyah-Singer index theorem. I always find that I give up because my analysis background is too weak -- most of the sources spend a lot of ...

**14**

votes

**8**answers

2k views

### Smooth classifying spaces?

Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular ...

**8**

votes

**6**answers

2k views

### What is an example of a topological space that is not homotopy equivalent to a CW-complex?

It would also be nice if someone can explain this comment appearing on the Wikipedia page on CW-complexes:
"The homotopy category of CW complexes is, in the opinion of some experts, the best if not ...

**3**

votes

**2**answers

368 views

### Legendrian homotopy of curves in a contact structure?

I'm aware of the great body of work on Legendrian knot theory in contact geometry, but suppose I'm curious just about homotopy and not isotopy. How does one understand the space of Legendrian loops ...

**1**

vote

**8**answers

2k views

### The core question of topology

As I see it, the core question of topology is to figure out whether a homeomorphism exists between two topological spaces.
To answer this question, one defines various properties of a space such as ...

**49**

votes

**9**answers

15k views

### Galois Groups vs. Fundamental Groups

In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of ...

**59**

votes

**9**answers

8k views

### understanding Steenrod squares

There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from ...

**11**

votes

**4**answers

1k views

### Elements of infinite order in a profinite group

Say G is a profinite group with elements of arbitrarily large order. Do elements of infinite order exist (A) if we assume G is abelian? (B) in general?
A start for (A): we can ask the same question ...

**15**

votes

**9**answers

2k views

### References for homotopy colimit

(1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...

**44**

votes

**5**answers

4k views

### Does homology have a coproduct?

Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor ...

**19**

votes

**3**answers

2k views

### What is a TMF in topology?

What is a topological modular form? How are they related to 'normal' (number-theoretic) modular forms?

**8**

votes

**2**answers

778 views

### Has anyone tabulated 2-knots? Would anyone like to try?

I'd love to have a list of 'small' 2-knots, for some sense of small. It's not clear what one should filter by, but there are two obvious candidates
Write a movie presentation, and count the frames.
...

**8**

votes

**5**answers

2k views

### Is the long line paracompact?

A manifold is usually defined as a second-countable hausdorff topological space which is locally homeomorphic to Rn. My understanding is that the reason "second-countable" is part of the definition is ...

**31**

votes

**6**answers

11k views

### How do you show that $S^{\infty}$ is contractible?

Here I mean the version with all but finitely many components zero.

**-1**

votes

**3**answers

3k views

**3**

votes

**3**answers

788 views

### Does the “continuous locus” of a function have any nice properties?

Suppose f:R→R is a function. Let S={x∈R|f is continuous at x}. Does S have any nice properties?
Here are some observations about what S could be:
S can be any closed set. For a closed set ...

**18**

votes

**4**answers

4k views

### Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?

An anonymous question from the 20-questions seminar:
Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?