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Let $\sum X$ be the reduce suspension of $X$.

Freudenthal Suspension Theorem: X is nondegenerately based and (n−1)-connected, where n ≥ 1. Then $\sum: \pi_q(X) \to \pi_q(\sum X)$ is a bijection if $q < 2n−1$ and a surjection if $q = 2n − 1$.

Note: nondegenerately based means the inclusion of the based point $\ast \hookrightarrow X $ is a cofibration.

We know that based on Freudenthal Suspension Theorem, for nondegenerately based path connected space X, $\sum X$ is simply connected.

My question:

Is there any example such that: X is path connected but not nondegenerately based and $\sum X$ is not simply connected.

co-co-nuts
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    As pointed out in comments at https://math.stackexchange.com/questions/195076/simply-connected-reduced-suspension-on-path-connected-x, Tom Goodwillie described an example: the unreduced cone on the set ${0, 1, 1/2, 1/3, 1/4, ...}$. See https://www.lehigh.edu/~dmd1/tg26, https://www.lehigh.edu/~dmd1/tg27, and https://www.lehigh.edu/~dmd1/tg28. – John Palmieri Apr 24 '20 at 02:59
  • @JohnPalmieri The OP looks for a path connected $X$. – Paul Frost Apr 24 '20 at 17:19
  • @PaulFrost Yes, and the cone on that set is path connected. – John Palmieri Apr 24 '20 at 17:29
  • @JohnPalmieri I should have read more carefully ;-) Perhaps you should give an official answer. – Paul Frost Apr 24 '20 at 17:32

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As pointed out in comments at math.stackexchange.com/questions/195076/…, Tom Goodwillie described an example: let $X$ be the unreduced cone $CZ$ on the set $Z=\{ 0,1,1/2,1/3,1/4,...\}$. See lehigh.edu/~dmd1/tg26, lehigh.edu/~dmd1/tg27, and lehigh.edu/~dmd1/tg28.

John Palmieri
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