# Questions tagged [at.algebraic-topology]

Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

**3**

votes

**1**answer

101 views

### Cohomogy of local systems over CW-complexes

Let $M$ be a finite CW-complex. Let $F$ be a finite rank local system over $M$ with coefficients in any field. Is it true that $\dim(H^k(M,F))$ is at most the number of $k$-cells times $\operatorname{...

**6**

votes

**1**answer

191 views

### Orientable with respect to complex cobordism?

I have learned that an orientation of a manifold $M$ with respect to ordinary cohomology is an ordinary orientation, that an orientation with respect to complex K-theory is a Spin$^c$ structure, and ...

**-1**

votes

**2**answers

119 views

### Directed colimit and homology

I am looking for a reference or a proof of the following fact:
Let $X_{1}\subset X_{2}\subset\dots $ be a sequence of (hausdorff) topological spaces indexed by natural numbers such that each $X_{i}\...

**3**

votes

**1**answer

114 views

### A pair of spaces equivalent to a pair of CW-complexes

Suppose that $X$ is a CW-complex and $Y$ a CW-subcomplex of $X$. Let $A$ be a closed subspace of $Z$ such that
$Z-A$ is homeomorhic to $X-Y$ and
$Z/A$ homeomorphic to $X/Y$ and
The closure of $Z-A$ ...

**1**

vote

**0**answers

210 views

### “a result of Atiyah implies that the top cell of an orientable manifold splits off stably”

I am looking for a reference for the following sentence
“a result of Atiyah implies that the top cell of an orientable
manifold splits off stably”
Thank you very much for helping me find a reference ...

**5**

votes

**0**answers

103 views

### Closed embedding of CW-complexes

Suppose that $i: X\rightarrow Y$ is a closed embedding such that $X$ and $Y $ are (retracts) of CW-complexes. Does it follow that $i$ is a cofibration ?
Remark: There is a similar question here, ...

**15**

votes

**2**answers

807 views

### What is the generator of $\pi_9(S^2)$?

This is more or less the same question as
[ What is the generator of $\pi_9(S^3)$? ], except what I would like to know is if it is possible to describe this map in a way
not only topologists can ...

**8**

votes

**1**answer

160 views

### Terminology about G- simplicial complexes

For a simplicial complex $X$ with an action of a discrete group $G$, we can impose the following condition, namely that if $g\in G$ stabilizes a given simplex $\sigma\subseteq X$, then $g:\sigma\to\...

**1**

vote

**0**answers

115 views

### Space of biholomorphic maps into a Riemann surface

Let $F$ be a Riemann surface and $Q\in F$. Consider $U:=(\mathbb{C}\cup\infty)\setminus [-1,1]^2$. I am interested in the space
$$X:=\{f:U\to F;\,\text{$f:U\to f(U)$ biholomorphic and $f(\infty)=Q$}\},...

**10**

votes

**0**answers

188 views

### Interpretation of determinants on commutative rings

In real Euclidian space, the result of the determinant can be interpreted as the oriented volume of the image of the unit cube under an invertible linear map.
This interpretation conceptually depends ...

**6**

votes

**0**answers

235 views

### Restriction of a cofibration to closed subspaces

Let $i: X\rightarrow Y$ be a cofibration between CW-complexes, more precisely a cellular embedding. Let $A$ be a closed subspace of $Y$ and $Z=i^{-1}(A)$. Let $$j: Z\rightarrow A$$ be the restriction ...

**5**

votes

**0**answers

83 views

### Analog of cellular approximation theorem for $CW_0$-complexes ($CW_\mathcal P$-complexes)

$CW_0$-complexes are analogs of $CW$-complexes, in which the "building blocks" are the rational disks $D^{n+1}_0$ whose boundaries are given by $\partial D^{n+1}_0= S^n_0$, where $S^n_0$ is a ...

**-1**

votes

**0**answers

87 views

### homotopy VS isotopy classes of embeddings

Let $X$ any compact set in $R^n$, not necessarily a manifold. Let $f,g:X \longrightarrow M^k $ be two homotopic PL embeddings of $X$. When $X$ is an $m$-manifold then the two embeddings are also ...

**11**

votes

**2**answers

381 views

### Realizing cohomology classes by submanifolds

In "Quelques propriétés globales des variétés différentiables", Thom gives conditions for a class in singular homology of a compact manifold to be realized by a smooth oriented submanifold (see e.g. ...

**2**

votes

**0**answers

74 views

### How much is the theory of piecewise-smooth manifolds different from the PL and smooth ones?

There are many differences between the smooth manifolds category and piecewise linear manifolds category when it comes to classification, embedding of these manifolds inside each other or otherwise. ...