From 518314a06c12c719ccaaa09c2dabd41e2c8e12f9 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sat, 16 May 2026 08:42:32 +0200 Subject: [PATCH 01/10] weakly contractible new --- properties/P000242.md | 21 +++++++++++++++++++++ theorems/T000890.md | 9 +++++++++ theorems/T000891.md | 9 +++++++++ theorems/T000892.md | 9 +++++++++ theorems/T000893.md | 9 +++++++++ 5 files changed, 57 insertions(+) create mode 100644 properties/P000242.md create mode 100644 theorems/T000890.md create mode 100644 theorems/T000891.md create mode 100644 theorems/T000892.md create mode 100644 theorems/T000893.md diff --git a/properties/P000242.md b/properties/P000242.md new file mode 100644 index 000000000..abb71a783 --- /dev/null +++ b/properties/P000242.md @@ -0,0 +1,21 @@ +--- +uid: P000241 +name: Weakly contractible +aliases: + - Homotopically trivial +refs: + - wikipedia: Weakly_contractible_space + name: Weakly contractible space on Wikipedia + - wikipedia: Homotopy_group + name: Homotopy group on Wikipedia + - zb: "1044.55001" + name: Algebraic Topology (Hatcher) +--- + +A nonempty, {P37} space for which all [homotopy groups](https://en.wikipedia.org/wiki/Homotopy_group) are trivial. Equivalently, the space is weakly homotopy equivalent to {S162}. + +---- +#### Meta-properties + +- This property is preserved by homotopy equivalences. +- This property is preserved by finite products. (See Proposition 4.2 in {{zb:1044.55001}}) diff --git a/theorems/T000890.md b/theorems/T000890.md new file mode 100644 index 000000000..c28f0b844 --- /dev/null +++ b/theorems/T000890.md @@ -0,0 +1,9 @@ +--- +uid: T000890 +if: + P000242: true +then: + P000037: true +--- + +By definition. diff --git a/theorems/T000891.md b/theorems/T000891.md new file mode 100644 index 000000000..5835cf2a2 --- /dev/null +++ b/theorems/T000891.md @@ -0,0 +1,9 @@ +--- +uid: T000891 +if: + P000242: true +then: + P000137: false +--- + +By definition. diff --git a/theorems/T000892.md b/theorems/T000892.md new file mode 100644 index 000000000..e6e0d8413 --- /dev/null +++ b/theorems/T000892.md @@ -0,0 +1,9 @@ +--- +uid: T000892 +if: + P000199: true +then: + P000242: false +--- + +Follows since any homotopy equivalence is also a weak homotopy equivalence. diff --git a/theorems/T000893.md b/theorems/T000893.md new file mode 100644 index 000000000..c325aecd5 --- /dev/null +++ b/theorems/T000893.md @@ -0,0 +1,9 @@ +--- +uid: T000893 +if: + P000242: true +then: + P000200: false +--- + +By definition, since the first homotopy group is the fundamental group. From f765ccda5718a0994cf9c91dcf44149ed1474734 Mon Sep 17 00:00:00 2001 From: Batixx Date: Sat, 16 May 2026 08:43:50 +0200 Subject: [PATCH 02/10] fix uid --- properties/P000242.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000242.md b/properties/P000242.md index abb71a783..b36168163 100644 --- a/properties/P000242.md +++ b/properties/P000242.md @@ -1,5 +1,5 @@ --- -uid: P000241 +uid: P000242 name: Weakly contractible aliases: - Homotopically trivial From 4644446f8a2772d6ae0632b501d98f614402c838 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Tue, 19 May 2026 04:39:32 +0200 Subject: [PATCH 03/10] Update properties/P000242.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000242.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/properties/P000242.md b/properties/P000242.md index b36168163..1c439e5d3 100644 --- a/properties/P000242.md +++ b/properties/P000242.md @@ -12,7 +12,7 @@ refs: name: Algebraic Topology (Hatcher) --- -A nonempty, {P37} space for which all [homotopy groups](https://en.wikipedia.org/wiki/Homotopy_group) are trivial. Equivalently, the space is weakly homotopy equivalent to {S162}. +A nonempty, {P37} space for which all [homotopy groups](https://en.wikipedia.org/wiki/Homotopy_group) $\pi_n(X)$ with $n\ge 1$ are trivial. Equivalently, the space is weakly homotopy equivalent to {S162}. ---- #### Meta-properties From e17c3ee78236ad00fb754557ba9853baaa3b9c8b Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Tue, 19 May 2026 06:50:26 +0200 Subject: [PATCH 04/10] Apply suggestion from @prabau Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- theorems/T000893.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000893.md b/theorems/T000893.md index c325aecd5..d19c30ab3 100644 --- a/theorems/T000893.md +++ b/theorems/T000893.md @@ -3,7 +3,7 @@ uid: T000893 if: P000242: true then: - P000200: false + P000200: true --- By definition, since the first homotopy group is the fundamental group. From 4c905bbac7404fd3439e8347675c78c931a82521 Mon Sep 17 00:00:00 2001 From: Batixx Date: Tue, 19 May 2026 11:08:02 +0200 Subject: [PATCH 05/10] fix refs --- properties/P000242.md | 12 +++++++----- 1 file changed, 7 insertions(+), 5 deletions(-) diff --git a/properties/P000242.md b/properties/P000242.md index 1c439e5d3..e89d32c77 100644 --- a/properties/P000242.md +++ b/properties/P000242.md @@ -6,13 +6,15 @@ aliases: refs: - wikipedia: Weakly_contractible_space name: Weakly contractible space on Wikipedia - - wikipedia: Homotopy_group - name: Homotopy group on Wikipedia - - zb: "1044.55001" - name: Algebraic Topology (Hatcher) + - wikipedia: Weak equivalence (homotopy theory) + name: Weak equivalence on Wikipedia + - zb: "0979.55010" + name: On the Universal Space for Group Actions with Compact Isotropy (Lück, Meintrup) --- -A nonempty, {P37} space for which all [homotopy groups](https://en.wikipedia.org/wiki/Homotopy_group) $\pi_n(X)$ with $n\ge 1$ are trivial. Equivalently, the space is weakly homotopy equivalent to {S162}. +A nonempty, {P37} space for which all homotopy groups $\pi_n(X)$ with $n\ge 1$ are trivial. Equivalently, the space is weakly homotopy equivalent to {S162}. + +Defined on page 2 in {{zb:0979.55010}}. ---- #### Meta-properties From 219324c7424db404c8ea03c79b7eb587d53d780b Mon Sep 17 00:00:00 2001 From: Batixx Date: Tue, 19 May 2026 11:11:15 +0200 Subject: [PATCH 06/10] add hatcher back (needed for products) --- properties/P000242.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/properties/P000242.md b/properties/P000242.md index e89d32c77..dc493801c 100644 --- a/properties/P000242.md +++ b/properties/P000242.md @@ -8,6 +8,8 @@ refs: name: Weakly contractible space on Wikipedia - wikipedia: Weak equivalence (homotopy theory) name: Weak equivalence on Wikipedia + - zb: "1044.55001" + name: Algebraic Topology (Hatcher) - zb: "0979.55010" name: On the Universal Space for Group Actions with Compact Isotropy (Lück, Meintrup) --- From 7ee2bff3fe1ff9bccdbb65792485ed875f59be63 Mon Sep 17 00:00:00 2001 From: Batixx Date: Tue, 19 May 2026 11:13:07 +0200 Subject: [PATCH 07/10] fix t892 --- theorems/T000892.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/theorems/T000892.md b/theorems/T000892.md index e6e0d8413..2789d708e 100644 --- a/theorems/T000892.md +++ b/theorems/T000892.md @@ -3,7 +3,7 @@ uid: T000892 if: P000199: true then: - P000242: false + P000242: true --- Follows since any homotopy equivalence is also a weak homotopy equivalence. From fde3377720aaf26be26912178b337f6fa8dbccd7 Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Thu, 21 May 2026 07:34:04 +0200 Subject: [PATCH 08/10] Update properties/P000242.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000242.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/properties/P000242.md b/properties/P000242.md index dc493801c..c0b2dbeac 100644 --- a/properties/P000242.md +++ b/properties/P000242.md @@ -14,7 +14,8 @@ refs: name: On the Universal Space for Group Actions with Compact Isotropy (Lück, Meintrup) --- -A nonempty, {P37} space for which all homotopy groups $\pi_n(X)$ with $n\ge 1$ are trivial. Equivalently, the space is weakly homotopy equivalent to {S162}. +$X$ is nonempty, {P37} and all [homotopy groups](https://en.wikipedia.org/wiki/Homotopy_group) $\pi_n(X)$ with $n\ge 1$ are trivial. +In other words, $X$ is [weakly homotopy equivalent](https://en.wikipedia.org/wiki/Weak_equivalence_(homotopy_theory)) to {S162}. Defined on page 2 in {{zb:0979.55010}}. From a9458969ccfe2562cbed6ff4307490ea3e1e56bc Mon Sep 17 00:00:00 2001 From: Felix Pernegger Date: Fri, 22 May 2026 06:30:39 +0200 Subject: [PATCH 09/10] Update properties/P000242.md Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> --- properties/P000242.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/properties/P000242.md b/properties/P000242.md index c0b2dbeac..4e6ef1364 100644 --- a/properties/P000242.md +++ b/properties/P000242.md @@ -14,6 +14,8 @@ refs: name: On the Universal Space for Group Actions with Compact Isotropy (Lück, Meintrup) --- +For each integer $n\ge 0$ every continuous map $S^n\to X$ is homotopic to a constant map. + $X$ is nonempty, {P37} and all [homotopy groups](https://en.wikipedia.org/wiki/Homotopy_group) $\pi_n(X)$ with $n\ge 1$ are trivial. In other words, $X$ is [weakly homotopy equivalent](https://en.wikipedia.org/wiki/Weak_equivalence_(homotopy_theory)) to {S162}. From 7c0174c11ee2a608c5a53de64426c66399c73ea4 Mon Sep 17 00:00:00 2001 From: Patrick Rabau <70125716+prabau@users.noreply.github.com> Date: Sat, 23 May 2026 19:04:04 -0400 Subject: [PATCH 10/10] more references and minor tidying up --- properties/P000242.md | 21 +++++++++++++++------ 1 file changed, 15 insertions(+), 6 deletions(-) diff --git a/properties/P000242.md b/properties/P000242.md index 4e6ef1364..fc08d7924 100644 --- a/properties/P000242.md +++ b/properties/P000242.md @@ -3,26 +3,35 @@ uid: P000242 name: Weakly contractible aliases: - Homotopically trivial + - $C^\infty$ refs: - wikipedia: Weakly_contractible_space name: Weakly contractible space on Wikipedia - - wikipedia: Weak equivalence (homotopy theory) - name: Weak equivalence on Wikipedia - zb: "1044.55001" name: Algebraic Topology (Hatcher) - zb: "0979.55010" - name: On the Universal Space for Group Actions with Compact Isotropy (Lück, Meintrup) + name: On the universal space for group actions with compact isotropy (Lück, Meintrup) + - zb: "1467.20042" + name: A stronger reformulation of Webb’s conjecture in terms of finite topological spaces (Piterman) + - zb: "0153.52905" + name: Theory of retracts (Borsuk) --- For each integer $n\ge 0$ every continuous map $S^n\to X$ is homotopic to a constant map. -$X$ is nonempty, {P37} and all [homotopy groups](https://en.wikipedia.org/wiki/Homotopy_group) $\pi_n(X)$ with $n\ge 1$ are trivial. +Equivalently, $X$ is nonempty, {P37} and all [homotopy groups](https://en.wikipedia.org/wiki/Homotopy_group) $\pi_n(X)$ with $n\ge 1$ are trivial. In other words, $X$ is [weakly homotopy equivalent](https://en.wikipedia.org/wiki/Weak_equivalence_(homotopy_theory)) to {S162}. -Defined on page 2 in {{zb:0979.55010}}. +Notation above: $S^n$ is the unit sphere in $\mathbb R^{n+1}$. + +Defined as *weakly contractible* for example on page 2 in {{zb:0979.55010}} +(accessible [here](https://www.academia.edu/53860578/On_the_universal_space_for_group_actions_with_compact_isotropy)). +Defined as *homotopically trivial* on page 281 of {{zb:1467.20042}}. +Borsuk uses the terminology $C^\infty$, defined as $X$ being *$n$-connected* for all $n\ge 0$ +(see page 30 of {{zb:0153.52905}}). ---- #### Meta-properties +- This property is preserved by finite products (see Proposition 4.2 in {{zb:1044.55001}}). - This property is preserved by homotopy equivalences. -- This property is preserved by finite products. (See Proposition 4.2 in {{zb:1044.55001}})