Can I file as a head of household without claiming an independent if my child has lived with me for over half a year because I let my ex claim our daughter?

To claim HOH you must have a dependent.

But you let your ex take the dependent.

@apiscanis 

According to the information on the IRS website, you can. (I assume you mean "dependent" where you said independent).

Here is a link to the IRS page: https://www.irs.gov/individuals/international-taxpayers/us-citizens-and-resident-aliens-abroad-head-of-household

As I understand it, if your unmarried daughter, who is not a foster daughter, lived with you more than half the calendar year and you paid more than half the household costs, you can claim HOH status - even though she is not your dependent.

(HOH = "head of household")

if you could claim the child as a dependent but allow the other parent to claim them you can still file as head of household. here are the rules from IRS PUB 17

Head of Household
1. You are unmarried or considered unmarried on the last day of the year. See Marital Status, earlier, and Considered Unmarried, later. 

2. You paid more than half of the cost of keeping up a home for the year.
3. A qualifying person lived with you in the home for more than half the year (except for temporary absences, such as school). .....

How to file. Indicate your choice of this filing status by checking the “Head of household” box
on the Filing Status line at the top of Form 1040 or 1040-SR. If the child who qualifies you for this filing status isn't claimed as your dependent in the Dependents section of Form 1040 or 1040-SR, enter the child's name in the entry space at the bottom of the Filing Status section. Use the Head of a household column of the Tax
Table, or Section D of the Tax Computation Worksheet, to figure your tax

If you are the custodial parent where the child physically lived for more than half the year (183 nights) then:

When you enter the dependent, you say that he/she is "Your child" (not you and your spouse if remarried), 
he/she lived with you the whole year, 
“no” the child did not pay more than half of his/her own support,
"yes", you have a custody agreement, 
and "yes", the other parent is claiming this year.  

That will give you the EIC, Child Care Credit and Head of Household filing status if you otherwise qualify.

The child would be listed as "non-dependent EIC & Dependent Care only".
The other (non-custodial) parent can claim the child’s exemption and child tax credit only and needs a signed 8332 form to do so.