# Tag Info

Accepted

### Is every based embedding from $S^{n}$ to $R^{n+1}\setminus\{{\bf 0}\}$ either the unit or a generator of $\pi_{n}(R^{n+1}\setminus\{{\bf 0}\})$?

I can solve $\frac{1}{2} + \epsilon$ of this problem. Edit: And now the solution is complete. Let $S \subset \mathbb R^{n+1} \setminus \{0\}$ be the image of an embedding of $S^{n}$. By the Jordan ...
• 103k
Accepted

### Spectra, Picard categories and their k-invariants

$A$ and $B$ are just $\pi_0$ and $\pi_1$; they’re citing a nontrivial but old theorem of stable homotopy theory in moving to maps out of $A\otimes \mathbb Z/2.$ I’d suggest reading an actual proof of ...
• 49.1k

### Is every based embedding from $S^{n}$ to $R^{n+1}\setminus\{{\bf 0}\}$ either the unit or a generator of $\pi_{n}(R^{n+1}\setminus\{{\bf 0}\})$?

Another way to do it is via homological degree, which generalizes the winding number argument. The Schonflies Theorem is a rather deep theorem, but I wanted to show that the question in the OP does ...
• 26.1k
Accepted

### Question on Proposition 2.22 of Hatcher's Algebraic Topology: How to induce a deformation retraction on the quotient space

All you have to know is that if $p : Y \to Z$ is a quotient map, then also $p \times id_I : Y \times I \to Z \times I$ is a quotient map. This is a well-known theorem from general topology. In ...
• 61.3k
Accepted

• 61.3k

### Let $A=\{(x,y) \in \Bbb S^n \times \Bbb S^n \mid x \ne y \}$. Let $f: \Bbb S^n \to A, \ x \mapsto(x,-x).$ Show that $f$ is a homotopy equivalence.

We have to find a homotopy inverse for $f$. It seems obvious that a nice candidate is $$g : A \to S^n, g(x,y) = x .$$ In fact $g \circ f = id$. The map $r = f \circ g$ is given by $$r(x,y) = (x,-x) .$$...
• 61.3k
1 vote

### $X=A\cup B$ be an open cover of $X$. If $X,A,B$ are simply connected , then $A\cap B$ path-connected?

Here is an interesting example to think about that doesn't prove or disprove the question. Let $X$ be the quasi-circle shown in the figure, a closed subspace of $\mathbb R^2$ consisting of a portion ...
• 505
1 vote
Accepted

### Cofibrant approximation of maps. [Hirschhorn 8.1.23]

Since $\tilde X$ is cofibrant and $\tilde g$ is a cofibration, the composition $0\to \tilde X\to E$ is a cofibration and $E$ is cofibrant.
• 49.1k
1 vote
Accepted

• 11k
1 vote

### How to show there is an $S^4$ included in a simplicial complex?

I do not believe that you can determine the lack of existence of a $\mathbb{Z}/2$-equivariant map $B \to S^3$ by knowing whether $B$ contains a "copy of $S^4$." For example, let $B$ consist ...
• 4,275
1 vote
Accepted

### A question related to the homotopy of two maps

A well-known result from algebraic topology is that any such $h$ can be deformed to an injective map. See cellular approximation theorem. Then $h$ (up to homotopy) can't be surjective, otherwise we'...
• 4,221
1 vote
Accepted

### Fundamental group of $S^2 \cup L$

Your first method is totally correct. However there are problems with your application of Van Kampen. Firstly you have to choose $U$ and $V$ such that $U\cup V=X$ (the space) and $U\cap V$ is path ...
• 8,482
1 vote
Accepted

• 1,479
1 vote

### Defining the sheaf of bigraded homotopy groups in motivic homotopy theory.

In classical homotopy theory, the stable homotopy category $\text{SH}$ is an additive category and so is its motivic analogue $\text{SH}(k)$ (they are even triangulated). One defines the group ...
• 304
1 vote

### Let $A=\{(x,y) \in \Bbb S^n \times \Bbb S^n \mid x \ne y \}$. Let $f: \Bbb S^n \to A, \ x \mapsto(x,-x).$ Show that $f$ is a homotopy equivalence.

Let V be a real vector space of dimension at least 2. Then VxV is the direct sum of the diagonal D, the set of points (x,x), and the antidiagonal A, the set of the points (x,-x). You should have no ...

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