Homology groups, homotopy equivalence and one point compactification

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December 1st 2010, 08:34 AM
math8

Homology groups, homotopy equivalence and one point compactification

Let $X= \textbf{R}^2 - \{ a,b \}$ and $Y = S^1 \times S^1 - \{c \}$ .

Do the spaces X and Y have the same homology groups? Are they homotopy equivalent? Are they Homeomorphic? Do they have homeomorphic one-point compactifications?

First, I was wondering if the space Y is homeomorphic to a cylinder? To me it looks like it's a Torus where one circle $S^1$ has been cut off. Or maybe, it is a Torus, where only point is missing?

I know that $H'_i (S^n) = \textbf{Z}$ if $i=n$ and $0$ otherwise. (Where $H'_i (X)$ is the i-dimensional reduced homology of the space $X$ )
Also I know as an application of the Mayer-Vietoris sequence, that If $S$ is a subspace of $S^n$ homeomorphic to $S^k$ for $0 \leq k < n$ , then $H'_i ( S^n -S) = \textbf{Z}$ if $i=n-k-1$ and $0$ otherwise.

Also, I know that $H'_i (X) = H_i (X$ ) for $i>0$ (Where $H_i(X)$ is the i-dimensional homology group of $X$ ) and that $H_0(X) = H'_0(X) \oplus \textbf{Z}$

But does any of this help answer the questions? Or is it totally irrelevant in this case? If this doesn't help, how do you go about answering the questions?
December 1st 2010, 06:12 PM
Drexel28

Quote:

Originally Posted by math8

Let $X= \textbf{R}^2 - \{ a,b \}$ and $Y = S^1 \times S^1 - \{c \}$ .

Do the spaces X and Y have the same homology groups? Are they homotopy equivalent? Are they Homeomorphic? Do they have homeomorphic one-point compactifications?

First, I was wondering if the space Y is homeomorphic to a cylinder? To me it looks like it's a Torus where one circle $S^1$ has been cut off. Or maybe, it is a Torus, where only point is missing?

I know that $H'_i (S^n) = \textbf{Z}$ if $i=n$ and $0$ otherwise. (Where $H'_i (X)$ is the i-dimensional reduced homology of the space $X$ )
Also I know as an application of the Mayer-Vietoris sequence, that If $S$ is a subspace of $S^n$ homeomorphic to $S^k$ for $0 \leq k < n$ , then $H'_i ( S^n -S) = \textbf{Z}$ if $i=n-k-1$ and $0$ otherwise.

Also, I know that $H'_i (X) = H_i (X$ ) for $i>0$ (Where $H_i(X)$ is the i-dimensional homology group of $X$ ) and that $H_0(X) = H'_0(X) \oplus \textbf{Z}$

But does any of this help answer the questions? Or is it totally irrelevant in this case? If this doesn't help, how do you go about answering the questions?

So you're ok with the alg. top. part? Assuming you mean $Y=\mathbb{S}^1\times\left(\mathbb{S}^1-\{c\}\right)$ then $X\not\approx Y$ . This is evident since $X$ isn't compact and $Y$ is. Also, I'm fairly sure (though don't quote me, I haven't checked it) that $X_{\infty}\not\approx Y_{\infty}$ because note that since $Y$ is already compact that $\{\infty\}$ is open (since $\{\infty\}=Y_{\infty}-\overbrace{Y}^{\text{compact}}$ and so $Y_{\infty}$ is not connected. That said, I'm fairly sure $X_{\infty}$ is.
December 1st 2010, 06:18 PM
Tinyboss

Y is homeomorphic to a cylinder if it's supposed to be S1 x (S1 - {c}), but I don't think that's how I'd read it, but rather as the torus minus one point. In the latter case, both spaces are homotopy equivalent to...what? Implying what about their homology groups?

Drexel28, I don't think Y is compact under the interpretation you mentioned.
December 1st 2010, 06:27 PM
Drexel28

Quote:

Originally Posted by Tinyboss

Y is homeomorphic to a cylinder if it's supposed to be S1 x (S1 - {c}), but I don't think that's how I'd read it, but rather as the torus minus one point. In the latter case, both spaces are homotopy equivalent to...what? Implying what about their homology groups?

Drexel28, I don't think Y is compact under the interpretation you mentioned.

Oops. That was careless haha.

Am I crazy, or is $X_{\infty}\approx Y$ ?
December 1st 2010, 06:46 PM
Tinyboss

I don't think so, since Y is not compact under either of the possible interpretations. I suspect it's meant to be the punctured torus and the twice-punctured plane, which are homotopy equivalent but not homeomorphic (since the one point compactification of the punctured torus is obviously the torus, but the 1pc of the twice-punctured plane is not a torus). Off the top of my head, I think the 1pc of the twice-punctured plane is probably homotopic to a wedge of three circles, but that's just a guess.
December 2nd 2010, 09:29 AM
math8

Thanks for your answer, and yes, you're right, Y is supposed to be the punctured Torus in this case.

I would say if X is homotopy equivalent to Y then, X and Y have isomorphic homology groups. But, how do we see they're homotopy equivalent? I read somewhere that the punctured Torus is Homotopy equivalent to the bouquet of two circles, why? Can we say that the twice punctured plane is also homotopy equivalent to the bouquet of two circles?

Also, If I understood well the reason why X and Y are not homeomorphic, is it because they have different one point compactifications? would you conclude in that case that their 1pc are non homeomorphic?
December 2nd 2010, 09:50 PM
xxp9

Let the rectangle $D=[-1,1]^2$ be the fundamental polygon of the torus $T^2$ , and let $\pi$ be the quotient map that identifies opposite edges, $\pi(D)=T^2$ .
The bouquet of two circles, E, is then homeomorphic to $\pi(\partial{D})$ , the image of the boundary of D.

Choose 0=(0, 0) to be the removed point. Along the radial beams you can define a deformation retract as follows. Given a point $p \in D$ other than 0, Let r(p) be the intersection of the beam 0p with $\partial{D}$ . Define $F: [0,1] \times T^2 \to T^2$ . $F(t,p)=(1-t)*p+t*r(p)$ . It's easy to verify that F is a strong deformation retract. This proves that the puctured torus is homotopy equivalent to the bouquet of two circles.

Since the plane is homemorphic to an open disk, you can define similarly a deformation retract, to show that a twice punctured open disk is homotopy equivalent to the bouquet of two circles.
December 3rd 2010, 04:38 AM
xxp9

A plane with 2 points deleted is homeomorphic to a sphere with 3 points deleted. So any small open neighborhood of $\infty$ of $X_{\infty}$ has 3 sheets, attached together by $\infty$ . While $Y_{\infty}$ is a torus, where there is an open neighborhood that is homeomorphic to $R^2$ for each point. This shows why $X_{\infty}$ is not homeomorphic to $Y_{\infty}$ . Neither for X and Y.

While $X_{\infty}$ and $Y_{\infty}$ have the same fundamental group $Z \times Z$ . To see it, draw a small triangle ABC on a sphere, it is easy to construct a homotopy so that the path ABC is equivalent to the path AC. Identify the 3 points A, B and C we get $X_{\infty}$ , with AB, BC and CA turning into circles. But there are only 2 generators( corresponding to the image of AB and BC) for the fundamental group of $X_{\infty}$ , but not 3, as Tinyboss guessed above.
December 4th 2010, 01:12 PM
math8

Thank you, that helps a lot!