# Tagged Questions

**1**

vote

**0**answers

102 views

### Existence of open dense subset in a Lie group

Let $G=S\cdot R$ be a Levi-Malcev decomposition of a connected complex Lie group
and $\Gamma$ a discrete subgroup of $G$ such that the subgroups
$\Gamma_g := \Gamma\cap (gSg^{-1})$ are finite for all ...

**7**

votes

**1**answer

301 views

### What are the applications of Dowker's theorem?

Let $R \subset X \times Y$ be any relation between sets $X$ and $Y$. CH Dowker constructed two simplicial complexes $K$ and $L$ associated to $R$:
a simplex in $K$ consists of finitely many elements ...

**6**

votes

**2**answers

520 views

### Computer aided homology computations

Background
I am currently working on the homology of some modulispace and there exists a much simpler chaincomplex with the same homology.
It is a quotient of a bisimplicial complex by a subcomplex. ...

**10**

votes

**0**answers

448 views

### Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and the Difficulty of Fundamental Groups

In manifold calculus, there are various analyticity estimates which run into trouble for codimension two embeddings. For instance, the functor $\operatorname{Emb}(M,N)$ is analytic in $M$ if $\dim M ...

**3**

votes

**1**answer

384 views

### Definition of Pontrjagin Classes

I am studying characteristic classes recently and find some quesions about Pontrjagin classes.
Firstly the definition of Pontrjagin classes is not that natural. When we talk about Pontrjagin classes ...

**9**

votes

**4**answers

892 views

### Coincidences amongst classifying spaces and Eilenberg Mac-Lane spaces

Given that $$\mathbb{R}P^{\infty} = B O(1) = K(\widehat{O(1)}, 1)$$ $$\mathbb{C} P^{\infty} = B U(1) = K( \widehat{U(1)}, 2)$$ is there any way to make sense of $$\mathbb{H}P^{\infty} = B Sp(1)$$ in a ...

**3**

votes

**2**answers

1k views

### How does singular homology H_n capture the number of n-dimensional “holes” in a space?

This is a foundational doubt I have. How does singular homology H_n capture the number of n-dimensional holes in a space?
We disregard the case of $H_0$ as it has the very satisfactory explanation ...

**3**

votes

**2**answers

426 views

### Euler characteristics and operator indices as exponents for Laurent polynomials

This question is rather vague. Are there any natural situations which involve Laurent polynomials of the form
$$\sum q^{a_i}\in\mathbb{Z}[q,q^{-1}]$$
where the $a_i$'s are either Euler characteristics ...

**5**

votes

**2**answers

513 views

### visualizing what's going on in based homotopy theory, et al.

I'm reading J.P. May's Concise Course in Algebraic Topology, and I'm having a lot of trouble visualizing how things work in Chapter 8, "Based cofiber and fiber sequences". Of course this is pretty ...

**19**

votes

**4**answers

2k views

### Visualizing how Cech cohomology detects holes

I think it's pretty intuitive how singular/simplicial cohomology detects "holes" in a space.
How can we directly visualize how and in what sense the Cech cohomology of a cover does this?
In ...