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Forum index » Science and Technology » Math » Undergraduate
Homotopy of maps
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Bruce Samuels
science forum beginner


Joined: 15 May 2005
Posts: 35

PostPosted: Sun May 21, 2006 10:40 pm    Post subject: Homotopy of maps Reply with quote
Could someone help me with the homotopy between these maps, I'm pretty sure
they are homotopic maps f,h:S^n->S^n where h(x)=f(x)+f(-x))/||f(x)+f(-x)||
(i.e. the norm)and f(-x)= -f(x) So how to show these maps are homotopic?
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Brian M. Scott
science forum Guru


Joined: 10 May 2005
Posts: 332

PostPosted: Sun May 21, 2006 11:01 pm    Post subject: Re: Homotopy of maps Reply with quote
On Sun, 21 May 2006 22:40:05 GMT, Bruce Samuels
<bsamuels@nyc.rr.com> wrote in
<news:9%5cg.9244$fj6.802@news-wrt-01.rdc-nyc.rr.com> in
alt.math.undergrad:

Quote:
Could someone help me with the homotopy between these maps, I'm pretty sure
they are homotopic maps f,h:S^n->S^n where h(x)=f(x)+f(-x))/||f(x)+f(-x)||
(i.e. the norm)and f(-x)= -f(x) So how to show these maps are homotopic?

You're starting with f and defining h in terms of it? If
so, h isn't always defined: what if f is the identity or the
antipodal map?

Brian
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W. Dale Hall
science forum Guru


Joined: 29 Apr 2005
Posts: 350

PostPosted: Mon May 22, 2006 1:01 am    Post subject: Re: Homotopy of maps Reply with quote
Brian M. Scott wrote:
Quote:
On Sun, 21 May 2006 22:40:05 GMT, Bruce Samuels
bsamuels@nyc.rr.com> wrote in
news:9%5cg.9244$fj6.802@news-wrt-01.rdc-nyc.rr.com> in
alt.math.undergrad:


Could someone help me with the homotopy between these maps, I'm pretty sure
they are homotopic maps f,h:S^n->S^n where h(x)=f(x)+f(-x))/||f(x)+f(-x)||
(i.e. the norm)and f(-x)= -f(x) So how to show these maps are homotopic?


You're starting with f and defining h in terms of it? If
so, h isn't always defined: what if f is the identity or the
antipodal map?

Brian

Um, not to mention the fact that his assumption regarding f(x)
(i.e., f(-x) = -f(x)) forces h to be undefined.

It's pretty difficult to show that a given map and one that
is undefined are homotopic. If you get a leg up in that problem,
be sure to let us know.

Dale.
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Brian M. Scott
science forum Guru


Joined: 10 May 2005
Posts: 332

PostPosted: Mon May 22, 2006 1:06 am    Post subject: Re: Homotopy of maps Reply with quote
On Mon, 22 May 2006 01:01:01 GMT, "W. Dale Hall"
<mailtowdunderscorehallatpacbelldotnet@last> wrote in
<news:h38cg.90498$dW3.12460@newssvr21.news.prodigy.com> in
alt.math.undergrad:

Quote:
Brian M. Scott wrote:

On Sun, 21 May 2006 22:40:05 GMT, Bruce Samuels
bsamuels@nyc.rr.com> wrote in
news:9%5cg.9244$fj6.802@news-wrt-01.rdc-nyc.rr.com> in
alt.math.undergrad:

Could someone help me with the homotopy between these maps, I'm pretty sure
they are homotopic maps f,h:S^n->S^n where h(x)=f(x)+f(-x))/||f(x)+f(-x)||
(i.e. the norm)and f(-x)= -f(x) So how to show these maps are homotopic?

You're starting with f and defining h in terms of it? If
so, h isn't always defined: what if f is the identity or the
antipodal map?

Um, not to mention the fact that his assumption regarding f(x)
(i.e., f(-x) = -f(x)) forces h to be undefined.

Ye gods. I didn't even notice that clause. Talk about
shooting yourself in the foot.

[...]

Brian
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Bruce Samuels
science forum beginner


Joined: 15 May 2005
Posts: 35

PostPosted: Mon May 22, 2006 2:09 am    Post subject: Re: Homotopy of maps Reply with quote
"Brian M. Scott" <b.scott@csuohio.edu> wrote in message
news:1e115tvp3w2mh$.xm429eegih80$.dlg@40tude.net...
Quote:
On Sun, 21 May 2006 22:40:05 GMT, Bruce Samuels
bsamuels@nyc.rr.com> wrote in
news:9%5cg.9244$fj6.802@news-wrt-01.rdc-nyc.rr.com> in
alt.math.undergrad:

Could someone help me with the homotopy between these maps, I'm pretty
sure
they are homotopic maps f,h:S^n->S^n where
h(x)=f(x)+f(-x))/||f(x)+f(-x)||
(i.e. the norm)and f(-x)= -f(x) So how to show these maps are homotopic?

You're starting with f and defining h in terms of it? If
so, h isn't always defined: what if f is the identity or the
antipodal map?

Brian

Ah yes, I left out an important assumption, that for all x in S^n, f(-x)
!= -f(x).
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Bruce Samuels
science forum beginner


Joined: 15 May 2005
Posts: 35

PostPosted: Mon May 22, 2006 2:39 am    Post subject: Re: Homotopy of maps Reply with quote
"Brian M. Scott" <b.scott@csuohio.edu> wrote in message
news:2uhmuupkzry9.465quxb63pfx$.dlg@40tude.net...
Quote:
On Mon, 22 May 2006 01:01:01 GMT, "W. Dale Hall"
mailtowdunderscorehallatpacbelldotnet@last> wrote in
news:h38cg.90498$dW3.12460@newssvr21.news.prodigy.com> in
alt.math.undergrad:

Brian M. Scott wrote:

On Sun, 21 May 2006 22:40:05 GMT, Bruce Samuels
bsamuels@nyc.rr.com> wrote in
news:9%5cg.9244$fj6.802@news-wrt-01.rdc-nyc.rr.com> in
alt.math.undergrad:

Could someone help me with the homotopy between these maps, I'm pretty
sure
they are homotopic maps f,h:S^n->S^n where
h(x)=f(x)+f(-x))/||f(x)+f(-x)||
(i.e. the norm)and f(-x)= -f(x) So how to show these maps are homotopic?

You're starting with f and defining h in terms of it? If
so, h isn't always defined: what if f is the identity or the
antipodal map?

Um, not to mention the fact that his assumption regarding f(x)
(i.e., f(-x) = -f(x)) forces h to be undefined.

Ye gods. I didn't even notice that clause. Talk about
shooting yourself in the foot.

[...]

Brian

Okay, I'm confusing the issue by not stating the original problem but
assuming I have a path to the solution. The problem reads If f:S^n->S^n has
odd degree, then f(-x) = -f(x) for some x in S^n. Then this suggestion is
offered.: Suppose not and consider h(x)=f(x)+f(-x))/||f(x)+f(-x)||
and use the fact that if f::S^n->S^n satisfies f(-x) = f(x) for all x in
S^n, then the degree f is even.

Bruce
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Brian M. Scott
science forum Guru


Joined: 10 May 2005
Posts: 332

PostPosted: Mon May 22, 2006 3:06 am    Post subject: Re: Homotopy of maps Reply with quote
On Mon, 22 May 2006 02:39:26 GMT, Bruce Samuels
<bsamuels@nyc.rr.com> wrote in
<news:yv9cg.3875$XR6.3439@news-wrt-01.rdc-nyc.rr.com> in
alt.math.undergrad:

[...]

Quote:
Okay, I'm confusing the issue by not stating the original
problem but assuming I have a path to the solution. The
problem reads If f:S^n->S^n has odd degree, then f(-x) =
-f(x) for some x in S^n. Then this suggestion is
offered.: Suppose not and consider
h(x)=f(x)+f(-x))/||f(x)+f(-x)||

Ah, okay; in that case of course h is well-defined.

Quote:
and use the fact that if
f::S^n->S^n satisfies f(-x) = f(x) for all x in S^n,
then the degree f is even.

There's an obvious thing to try:

H(x, t) = [f(x) + t*f(-x)]/||f(x) + t*f(-x)||

Have you tried to show that it does (or doesn't) work?

Brian
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