# Tagged Questions

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

228 views

### Smith normal form of graded modules. (Major edit)

Ok, this is a major rewriting of my previous entry which no one answered. Let us have two graded $F[t]$-modules M and N with bases $m_1, \ldots, m_m$ and $n_1, \ldots, n_n$, respectively, and $F$ is ...
170 views

36 views

### Source request for $H^*(B\mathrm{TOP},\mathbb{Q})\cong H^*(BO,\mathbb{Q})$

Let $B\mathrm{TOP}$ denote the classifying space for microbundles, i.e. $B\operatorname{Homeo}(\mathbb{R}^n,0)$. Now we get a map from $BO$ to $B\mathrm{TOP}$ via the inclusion. Let $f$ denote the ...
53 views

### Homotopy of boundary paths

Let $G$ be a bounded, simply connected, open set in $\mathbb{R}^2$, and let $\gamma_1$ and $\gamma_1$ denote two paths such that $\gamma_0(0)=\gamma_1(0)$, $\gamma_0(1)=\gamma_1(1)$, and the following ...
34 views

### If $M = \partial W$, with $W$ parallelizable, then an embedding $\iota : M \to S^{n+k}$ extends to an embedding $W \to D^{n+k+1}$

Suppose $M$ is an $n$-dimensional $s$-parallelizable manifold which is the boundary of the parallelizable compact manifold $W$. It is claimed in Milnor & Kervaire's Groups of Homotopy Spheres ...
102 views

### Surjectivity of Edge morphism in A-H cohomology Spectral Sequence

As the title suggests, I'm interested in proving the following claim: Recall the AH-spectral sequence:$$E_2^{pq}=H^p(X,\mathcal{H}^q(\ast)) \Longrightarrow \mathcal{H}^{p+q}(X)$$ and since ...
50 views

### Examples of calculating perverse sheaves on algebraic varieties with easy stratification.

This question is also asked in mathoverflow http://mathoverflow.net/questions/232589/examples-of-calculating-perverse-sheaves-on-algebraic-varieties-with-easy-strati I have been learning intersection ...
80 views

### Lie-group existence on universal covering manifold

Let $X$ be an n-dimensional smooth manifold with Lie group $G$ acting transitively on $X$, i.e. $X$ is a homogeneous space. Let $\tilde{X}$ be the associated universal covering space. To what extent ...
45 views

### Characterisation of Stable Cohomological Operations: $\Sigma (\tau_n (\imath_{A,n}))=\rho_{n+1}^*(\tau_{n+1}(\imath_{A,n+1}))$

I've started studying (stable) cohomological operations on my lecture notes, and I was given that an equivalent definition for a family of cohomological operations to be a stable cohomological ...
129 views

### “Global” dimension for topological spaces // Geometric interpretation for global dimension of rings

The global dimension of a ring $R$ is the supremum of the projective dimensions of it's $R$-modules. $$\dim (R)=\sup \{\dim_\mathrm{proj}(M):M \in R\text{-mod} \}$$ I'd like to have some geometric ...
108 views

### Learning Galois theory geometrically?

Recently I started poking at algebraic geometry and commutative algebra. My background is basic category theory and basic algebraic topology. I don't know a lot of other mathematics. I noticed Galois ...
59 views

### Curves Knotted in the Torus

I've been struggling with the following problem. I suspect it's actually not so hard, but I'm missing something relatively obvious about the proof. The attached image will help. Suppose $K$ and $L$ ...
I'm trying to compute the homology of $X = (I \times I)/\sim$, where $(0,0)\sim (0,1) \sim (1,0) \sim (1,1)$. I want to do this via cellular homology, using degrees, etc, but I don't got that very ...