# 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.

32 views

21 views

### Does weak Hausdorffication preserve equalizers and finite products?

There is a "weak Hausdorffication"-functor $wh$ from the category of compactly generated spaces (CG) to the category of compactly generated weak Hausdorff spaces (CGWH) given by quotienting out the ...
123 views

### Is $(\#^k \Bbb{RP}^2) \times I$ an $\mathbb{RP}^2$-irreducible 3-manifold?

Consider $S$ a surface homeomorphic to a connected sum of $n$ projective planes, $n \geq 2$. Can there be a two sided projective plane embedded in $[-\epsilon,\epsilon]\times S$?
51 views

### An algebra problem from spectral sequence [duplicate]

Recently, I am reading the article "You Could Have Invented Spectral Sequences" by Timothy Y. Chow. Link: http://www-math.mit.edu/~tchow/spectral.pdf In page 17, he used the following splitting which ...
110 views

### Triangulation of Torus in three different ways, but two of them are wrong.

In my geometry notes the writer states that the following two, are not triangulations for the torus: On the contrary this is a good triangulation: I tried to wrap a piece of paper in order the ...
83 views

### Why do Wikipedia and nLab seem to give completely incompatible definitions of the term “simplex”?

nLab seems to define that a simplex is an inhabited finite totally-ordered set. Wikipedia seems to define that simplex is a subset of real affine space satisfying certain conditions. Q. What's ...
27 views

### first homolgy group of a disk with $n$ holes

Let $D^2$ be a 2-dim disk with $n$ holes, i.e $D^2\setminus(S^0 * D^2)^n$. Then is it true that the first homology group of this space is $\mathbb{Z}^n$.
38 views

### Why this cocycle $\mathrm{char} (h)$ is not a coboundary?

Maybe this is a stupid question and I'm missing something very trivial. Let $X$ be a smooth manifold, $$h \colon Z_{k-1} (X, \mathbb{Z}) \rightarrow \mathbb{R}/\mathbb{Z}$$ an abelian group morphism ...
42 views

### Quotient homeomorphic to product

$X_1, X_2$ are topological spaces and $G_1,G_2$ are groups acting freely and properly discontinuously on these spaces by homeomorphisms. (Means that for every $g\in G$ the map $(g,x) \mapsto g(x)$ is ...
100 views

### Classify Open Sets in $\mathbb R^2$

In $\mathbb R$, we know that connected open set is $(0,1)$ under homeomorphism. I am wondering what is the situation in $\mathbb R^2$. From $\mathbb R^2-\text{pt}\simeq S^1$, we will have two open ...
27 views

### Homology of manifolds using submanifolds [duplicate]

Motivation I have learned algebraic topology. In simplical homology, we define $C_k(X)$ as an abelian group freely generated by $k$-dimensional skeleton $X^{(k)}$, and boundary operator $\partial_k$ ...
97 views

### Nontrivial cup product realized in $\Bbb R^4$

Let $A$ be a closed subspace $A$ of $[0,1]^4$---let's say, a subcomplex of some triangulation of the cube. I would like to show that the cup product $H^2(A)\times H^2(A)\to H^4(A)$ is trivial (or at ...
110 views

### Maps from Sum of Projective Planes to Circle

this is a problem from Lee's Topological Manifolds, 11-21. It asks the following: What compact, connected surfaces $M$ admit a non-nullhomotopic map to the circle? So, we use the classification of ...
49 views

### compact oriented $4k$-manifold-> euler characteristic is congruent to the signature mod $2$

Let $M$ be a compact oriented $4k$-manifold, $\chi_M$ the euler characteristic of $M$ and $sig(M)$ the signature of $M$. Why is $\chi_M \equiv sig(M)$ mod $2$? In the book " a concise course in ...
28 views

### Why is the sectional shape of a simply connected, oriented 4-manifold an isomorphism?

Let $M$ be a simply connected, compact and $\mathbb{Z}$-oriented 4-dimensional manifold. Let $\mu\in H_4(M;\mathbb{Z})$ be a fundamental class of $M$ (here is $H_4(M;\mathbb{Z})$ the singular homology ...
33 views

### Does weak Hausdorffication preserve inclusions?

There is a "weak Hausdorffication"-functor $wh$ from the category of compactly generated spaces (CG) to the category of compactly generated weak Hausdorff spaces (CGWH) given by quotienting out the ...
95 views

### Homology of sphere-complements

I have to solve the following questions: "For a subset $X \subset S^n$ determine the homology group $H_i(S^n - X)$, where (a) $X \cong S^l \vee S^k$ (b) $X \cong S^l \sqcup S^k$ (disjoint union) " ...
60 views

### $f(z) = z^3 + 2z + 7$. Calculate $f_* : H_2\to H_2$.

Let $f(z) = z^3 + 2z + 7$. $f: \hat{\mathbb{C}} \to \hat{\mathbb{C}}$, with $f(\infty) = \infty$. Calculate $f_* : H_2(\hat{\mathbb{C}}, \mathbb{Z}) \to H_2(\hat{\mathbb{C}}, \mathbb{Z})$. What I ...
37 views

60 views

### Reference for a vector space lemma of Hopf?

I've been told that the following is due to Hopf. Let $A, B, C$ be complex vector spaces. Given any linear map $$v:A\otimes B \rightarrow C,$$ where $A, B, C$ are complex vector spaces and $v$ is ...
44 views

### Hatcher Corollary 4.12

Just to ask a quick question regarding a corollary 4.12 in Hatcher: "A CW pair $(X,A)$ is n-connected if all the cells in $X-A$ have dimension greater than $n$. In particularly the pair $(X,X^n)$ is n-...
67 views

### Zeroth homotopy group: what exactly is it?

What are the elements in the zeroth homotopy group? Also, why does $\pi_0(X)=0$ imply that the space is path-connected? Thanks for the help. I find that zeroth homotopy groups are rarely discussed in ...
160 views

### Getting the most general form of Mayer-Vietoris from the axioms of homology

I'd like to derive the most general form of the Mayer-Vietoris sequence from the Eilenberg-Steenrod axioms for homology (in particular: I do not want to use the definition of $H_\ast(X)$ in terms of ...
131 views

### The fundamental group of $\mathbb{R}^3$ with its non-negative half-axes removed

Determine whether the fundamental group of $\mathbb{R}^3$ with its non-negative half-axes removed is trivial, infinite cyclic, or isomorphic to the figure eight space. I found this answer: Why ...
54 views

### Equivariant cohomology via equivariant sheaves

Ordinary cohomology of topological space $X$ are known to be the cohomology of constant sheaf. Question Is there analogous description for equivariant cohomology? More precisely. Consider category ...
60 views

### Showing $\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}/(1,-1,0)\mathbb{Z}+(0,1,1)\mathbb{Z}+(1,0,-1)\mathbb{Z}\cong \mathbb{Z}$

I have to prove that if $V_K = \{v_0, v_1, v_2\}$ and $K = \{\{v_0\}, \{v_1\}, \{v_2\}, \{v_0, v_1\}, \{v_0, v_2\}, \{v_1, v_2\}\}$ then $H_q(K, \mathbb{Z})\cong \mathbb{Z}$ for $q = 0, 1$. Already ...
30 views

### Why did $Ext$ appear to make the sequence exact after taking its dual?

The above question is about the exact sequence in the bottom of the following figure from p196 of Hatcher's text. After taking the dual of the original short exact sequence, $Ext$ comes in at the end ...
87 views

### Counterexamples on homotopy equivalence and infinite product [closed]

Let $(X,a),(Y,b)$ be pointed spaces and $f:(X,a)\rightarrow (Y,b)$ be a continuous function. If the natural homomorphism $f_*:\pi_1(X,a)\rightarrow \pi_1(Y,b)$ is a group isomorphism, is $f$ ...
87 views

### Interpretation for the curvature and monodromy of a connection - Reality check

Let $P \to M$ be a principal $G$-bundle with connection form $\omega \in \Omega^1(P,\mathfrak{g})$. Here are the statements I'm basing my viewpoint on: A connection is flat (vanishing curvature)...
In exercise 1.3.28 in Hatcher's Algebraic Topology, we are asked to show that for a covering space action of a group $G$ on a simply-connected space $Y$, $\pi_1(Y/G)$ is isomorphic to $G$. This is a ...
### Is $X$ a subset of $CX$?
In Spanier's, Algebraic Topology, he writes: "A topological pair $(X,A)$ consists of a topological space $X$ and a subspace $A \subset X$." In a question at the end of the section he asks a ...