Relation between transport functor of a fibration and a Hurewicz connection on it
5
$begingroup$
Let $Aoverset{alpha}{rightarrow}B$ be a (Hurewicz) fibration.
- The homotopy lifting property w.r.t a fiber $alpha ^{-1}(b)$
furnishes for each path $bto b^prime$ in the base a continuous map
$alpha ^{-1}(b)to alpha ^{-1}(b^prime)$. Moreover, this
assignment extends to a functor $pi_1Blongrightarrow
mathsf{hTop}$. - On the other hand, as a fibration $begin{smallmatrix}A\downarrow\Bend{smallmatrix}$ admits a Hurewicz connection $s$. Given such a connection it is tempting to send a path $boverset{gamma}{to} b^prime $ in the base to the following set function (analogously to covering space theory) $$alpha^{-1}(b)longrightarrow alpha ^{-1}(b^prime),quad amapsto operatorname{eval}_1s(a,gamma).$$ I suspect this set function might be continuous, but I see no reason for it to be a homotopy equivalence, since $s(a,gamma)$ need not be related in a nice way to lifts of opposite path $boverset{bargamma}{leftarrow} b^prime $.
Questions.
- Is fiber transport along a Hurewicz connection continuous?
- (Assuming continuity) Is fiber transport along a Hurewicz connection functorial?
- (Assuming continuity) Does it coincide with the first transfer functor?
- Suppose the fibers are all homeomorphic. Are there any interesting conditions that make the transport functor $pi_1Blongrightarrow mathsf{hTop}$ lift to $mathsf{Top}$? That is, can we obtain such homeomorphisms via transport?
I thought about possible functoriality of the transport along the Hurewicz connection. We want $operatorname{eval}_1(a,delta ast gamma)=operatorname{eval}_1s(operatorname{eval}_1s(a,gamma),delta,)$. We may consider the concatenation $s(operatorname{eval}_1s(a,gamma),delta)ast s(a,gamma)$ which seems to lift $delta ast gamma$, but I'm not quite sure where to go from here.
general-topology algebraic-topology homotopy-theory fibration
asked Nov 11 '18 at 22:01
ArrowArrow
5,19931446
$endgroup$
add a comment |
5
$begingroup$
Let $Aoverset{alpha}{rightarrow}B$ be a (Hurewicz) fibration.
- The homotopy lifting property w.r.t a fiber $alpha ^{-1}(b)$
furnishes for each path $bto b^prime$ in the base a continuous map
$alpha ^{-1}(b)to alpha ^{-1}(b^prime)$. Moreover, this
assignment extends to a functor $pi_1Blongrightarrow
mathsf{hTop}$. - On the other hand, as a fibration $begin{smallmatrix}A\downarrow\Bend{smallmatrix}$ admits a Hurewicz connection $s$. Given such a connection it is tempting to send a path $boverset{gamma}{to} b^prime $ in the base to the following set function (analogously to covering space theory) $$alpha^{-1}(b)longrightarrow alpha ^{-1}(b^prime),quad amapsto operatorname{eval}_1s(a,gamma).$$ I suspect this set function might be continuous, but I see no reason for it to be a homotopy equivalence, since $s(a,gamma)$ need not be related in a nice way to lifts of opposite path $boverset{bargamma}{leftarrow} b^prime $.
Questions.
- Is fiber transport along a Hurewicz connection continuous?
- (Assuming continuity) Is fiber transport along a Hurewicz connection functorial?
- (Assuming continuity) Does it coincide with the first transfer functor?
- Suppose the fibers are all homeomorphic. Are there any interesting conditions that make the transport functor $pi_1Blongrightarrow mathsf{hTop}$ lift to $mathsf{Top}$? That is, can we obtain such homeomorphisms via transport?
I thought about possible functoriality of the transport along the Hurewicz connection. We want $operatorname{eval}_1(a,delta ast gamma)=operatorname{eval}_1s(operatorname{eval}_1s(a,gamma),delta,)$. We may consider the concatenation $s(operatorname{eval}_1s(a,gamma),delta)ast s(a,gamma)$ which seems to lift $delta ast gamma$, but I'm not quite sure where to go from here.
general-topology algebraic-topology homotopy-theory fibration
asked Nov 11 '18 at 22:01
ArrowArrow
5,19931446
$endgroup$
add a comment |
5
5
5
$begingroup$
Let $Aoverset{alpha}{rightarrow}B$ be a (Hurewicz) fibration.
- The homotopy lifting property w.r.t a fiber $alpha ^{-1}(b)$
furnishes for each path $bto b^prime$ in the base a continuous map
$alpha ^{-1}(b)to alpha ^{-1}(b^prime)$. Moreover, this
assignment extends to a functor $pi_1Blongrightarrow
mathsf{hTop}$. - On the other hand, as a fibration $begin{smallmatrix}A\downarrow\Bend{smallmatrix}$ admits a Hurewicz connection $s$. Given such a connection it is tempting to send a path $boverset{gamma}{to} b^prime $ in the base to the following set function (analogously to covering space theory) $$alpha^{-1}(b)longrightarrow alpha ^{-1}(b^prime),quad amapsto operatorname{eval}_1s(a,gamma).$$ I suspect this set function might be continuous, but I see no reason for it to be a homotopy equivalence, since $s(a,gamma)$ need not be related in a nice way to lifts of opposite path $boverset{bargamma}{leftarrow} b^prime $.
Questions.
- Is fiber transport along a Hurewicz connection continuous?
- (Assuming continuity) Is fiber transport along a Hurewicz connection functorial?
- (Assuming continuity) Does it coincide with the first transfer functor?
- Suppose the fibers are all homeomorphic. Are there any interesting conditions that make the transport functor $pi_1Blongrightarrow mathsf{hTop}$ lift to $mathsf{Top}$? That is, can we obtain such homeomorphisms via transport?
I thought about possible functoriality of the transport along the Hurewicz connection. We want $operatorname{eval}_1(a,delta ast gamma)=operatorname{eval}_1s(operatorname{eval}_1s(a,gamma),delta,)$. We may consider the concatenation $s(operatorname{eval}_1s(a,gamma),delta)ast s(a,gamma)$ which seems to lift $delta ast gamma$, but I'm not quite sure where to go from here.
general-topology algebraic-topology homotopy-theory fibration
asked Nov 11 '18 at 22:01
ArrowArrow
5,19931446
$endgroup$
Let $Aoverset{alpha}{rightarrow}B$ be a (Hurewicz) fibration.
- The homotopy lifting property w.r.t a fiber $alpha ^{-1}(b)$
furnishes for each path $bto b^prime$ in the base a continuous map
$alpha ^{-1}(b)to alpha ^{-1}(b^prime)$. Moreover, this
assignment extends to a functor $pi_1Blongrightarrow
mathsf{hTop}$. - On the other hand, as a fibration $begin{smallmatrix}A\downarrow\Bend{smallmatrix}$ admits a Hurewicz connection $s$. Given such a connection it is tempting to send a path $boverset{gamma}{to} b^prime $ in the base to the following set function (analogously to covering space theory) $$alpha^{-1}(b)longrightarrow alpha ^{-1}(b^prime),quad amapsto operatorname{eval}_1s(a,gamma).$$ I suspect this set function might be continuous, but I see no reason for it to be a homotopy equivalence, since $s(a,gamma)$ need not be related in a nice way to lifts of opposite path $boverset{bargamma}{leftarrow} b^prime $.
Questions.
- Is fiber transport along a Hurewicz connection continuous?
- (Assuming continuity) Is fiber transport along a Hurewicz connection functorial?
- (Assuming continuity) Does it coincide with the first transfer functor?
- Suppose the fibers are all homeomorphic. Are there any interesting conditions that make the transport functor $pi_1Blongrightarrow mathsf{hTop}$ lift to $mathsf{Top}$? That is, can we obtain such homeomorphisms via transport?
I thought about possible functoriality of the transport along the Hurewicz connection. We want $operatorname{eval}_1(a,delta ast gamma)=operatorname{eval}_1s(operatorname{eval}_1s(a,gamma),delta,)$. We may consider the concatenation $s(operatorname{eval}_1s(a,gamma),delta)ast s(a,gamma)$ which seems to lift $delta ast gamma$, but I'm not quite sure where to go from here.
general-topology algebraic-topology homotopy-theory fibration
general-topology algebraic-topology homotopy-theory fibration
asked Nov 11 '18 at 22:01
ArrowArrow
5,19931446
asked Nov 11 '18 at 22:01
ArrowArrow
5,19931446
edited Nov 11 '18 at 22:08
Arrow
asked Nov 11 '18 at 22:01
ArrowArrow
5,19931446
asked Nov 11 '18 at 22:01
ArrowArrow
5,19931446
asked Nov 11 '18 at 22:01
ArrowArrow
5,19931446
5,19931446
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
draft saved
draft discarded
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2994508%2frelation-between-transport-functor-of-a-fibration-and-a-hurewicz-connection-on-i%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
draft saved
draft discarded
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
draft saved
draft discarded
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2994508%2frelation-between-transport-functor-of-a-fibration-and-a-hurewicz-connection-on-i%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
