5
votes
$n-$th singular homology group of a space with path connected components
You say that you can see that each $C_n(X) \cong \bigoplus_\alpha C_n(X_\alpha)$. Can you see that these isomorphisms are compatible with the chain complex structure? That is, write $\varphi_n : C_n(X)...
- 31.4k
5
votes
Is there a notion of homology equivalence of spaces, similar to how there is homotopy equivalence?
I realise that one could define two spaces to be "homology equivalent" if they have the same homology, as people say, i.e., if they have all homology groups isomorphic.
Yeah, such notion ...
- 36.6k
4
votes
Accepted
Is there a vector bundle of rank $\geq2$ admitting no nonzero proper vector sub-bundles?
Since $H^1(S^2, \mathbb{Z}_2) = 0$, any real line bundle over $S^2$ is trivial and thus admits a nowhere-vanishing section. But $TS^2$ does not (because, e.g., $e(TS^2) = \chi(TS^2)$ is nonzero), so $...
- 24.1k
3
votes
Lens space $L^k(n) \equiv S^{2k-1}/\mathbb{Z}_n$: Stiefel-Whitney class and non/spin manifold
Since $M$ is orientable, we have $w_1(TM)=0$ for all $k,n$.
We know that $w_2(TM)\in H^2(M;\Bbb Z_2)$, and explicit computation shows that when $n$ is odd, we have $H^i(M;\Bbb Z_2)\cong 0$ for $1\le i\...
- 3,059
3
votes
Degree of maps $S^n \to S^n$ using local degree formula - practicality of determining signs
Here's an answer that works in the smooth case. That is, we consider $X$ and $Y$ as smooth manifolds, and we assume $f:X\rightarrow Y$ is smooth.
Choose orientations on $X$ and $Y$. For topological ...
- 48.8k
3
votes
Poincaré duality and coefficients in the circle group
Poincaré duality does not actually require a ring structure on the coefficients. Suppose $R$ is a ring and $M$ is an $R$-module. The scalar multiplication of $M$ then gives a cap product operation $...
- 312k
2
votes
Accepted
Simplifying a quotient group $\frac{\mathbb Z(\{e_1,e_2,...,e_{n+1}\})}{\mathbb Z(\{e_1,2e_2-e_1,...,2e_{n+1}-e_n\})}$
Let $f_k:=e_k-2^{n-k+1}e_{n+1}$ for all $k: 1\le k\le n,$ and $f_0:=e_1-f_1=2^ne_{n+1}.$ Our quotient group is
$$\frac{\Bbb Z(\{f_1,\dots,f_n,e_{n+1}\})}{\Bbb Z(\{f_0,f_1,\dots,f_n\})}\cong\frac{\Bbb ...
- 18.3k
2
votes
Simplifying a quotient group $\frac{\mathbb Z(\{e_1,e_2,...,e_{n+1}\})}{\mathbb Z(\{e_1,2e_2-e_1,...,2e_{n+1}-e_n\})}$
A systematic way to solve this kind of problems, is by writing it as the cokernel of the map
$$\Phi:\mathbb Z^{n+1}\to\mathbb Z^{n+1}$$
where the matrix representation of $\Phi$ in the basis $e_1,\...
- 3,574
2
votes
Sections form a vector space
Your approach is correct. For any vector bundle, you correctly defined the sum of two sections, but concerning continuity of the sum you have indeed a gap.
I suggest to proceed as as follows.
...
- 3,355
2
votes
Accepted
Proof that $Aut(P) \cong H^2(X;\mathbb Z)$, where $P \to X$ is a principal $PU(\mathcal H)$-bundle.
As Ben pointed out in a comment, Atiyah and Segal actually have this result in the paper as Proposition 2.2.
Their proof is quite terse, so I thought I would at least fill in the details here. The ...
- 353
2
votes
Degree of maps $S^n \to S^n$ using local degree formula - practicality of determining signs
You are right, Hatcher is a bit sloppy when defining the local degree of a map $f : S^n \to S^n$. Based on the diagram on p. 136 he writes
Via these four isomorphisms, the top two groups in the
...
- 67.2k
2
votes
Accepted
On the definition of homotopy extension property
With the additional assumption that $r$ in Definition 2 needs to be a retraction, both definitions are equivalent:
Given $H: A \times I \to T$ and $f: X \to T$ such that $H(-,0) = f|_A$ allows us to ...
- 9,272
2
votes
Accepted
Understanding Wikipedia's definition of the "join" of topological spaces
The join of two topological spaces is space that inherits some of the properties of each space, which somehow interpolates from one to the other as the parameter $t$ varies along $[0, 1]$. The most ...
- 21k
2
votes
Accepted
Dense algebraic curves
The trivial answer is yes: If all the $a_{ij}$ are zero, then $A = \mathbb R^2$ is dense.
But if at least one of the $a_{ij} \neq 0$, then $A$ is not dense:
The function $f(x,y) = \sum_{i,j} a_{ij} x^...
- 6,071
2
votes
Accepted
How to make associated ribbon surface?
First, on your hand drawn graph, you have the red arrows reversed in one segment on the right side. After you fix that the glueing works as follows.
Close off each ribbon with a small line segment. ...
- 5,258
1
vote
Accepted
If a submanifold is homeomorphic to the ambient manifold but is a proper subset, then the inclusion map can't be proper
Your argument looks correct to me. Note that depending on the definition of submanifolds, you may need to add the assumption that $N$ does not have boundary.
In fact an inclusion with locally compact ...
- 7,762
1
vote
What is the Geometric Interpretation of the Addition of n-Singular Simplices?
You did not miss anything: there's no particular geometric intuition, it's really just algebraic formalities. A chain is just a "vector" assigning a numerical coordinate to each singular ...
- 109k
1
vote
What is the connection between the two definitions of homology in 1.1 of Rotman's An Introduction to Homological Algebra?
Don't worry, Rotman just wants to give examples of the use of the term "homology" in mathematics. It should be a motivational introduction, but I am afraid that it is not really a felicitous ...
- 67.2k
1
vote
Is there a symmetric embedding of the torus in $\mathbb{R}^3$?
If the embedding is locally the standard embedding of the $2$-disk in the $3$-disk then we can define orientations using local homology and follow ideas from the comments to the question. Since $s$ ...
- 7,762
1
vote
Existence of a cone neighbourhood in an open disc with a point in the border
If the authors meant anything else by "cone neighborhood" than a neighborhood of $a$ that is also a cone, they surely would have given a definition for the phrase. You will have to check ...
- 40.6k
1
vote
Existence of a cone neighbourhood in an open disc with a point in the border
By cone neighbourhood in P do they mean a cone $aL$ such that $aL\subset P$ is true and also there is an open subset $O\subset P$ such that $a\in O\subset aL$?
Yes, a cone neighborhood of $a \in P$ ...
- 67.2k
1
vote
Understanding why collapsing $CX$ and $CY$ gives us $\sum (X \times Y)$?
This is an extra-long proof of $1$. My hope is that after reading it thoroughly several times, you'll be equipped to answer question 2 and 3 yourself.
The definition of the (reduced) join $(X,x_0)\...
- 48.8k
Only top scored, non community-wiki answers of a minimum length are eligible